A STUDY OF THE STABILEZING AUTOMORPHlSMS OF A FlNlTE GROUP ’ihesis fies" the Degree ofPh. 5, MECEfiGAE-‘é S'WE WWERSITY BRUGE STAAE. 1975 This is to certify that the thesis entitled A Study of the Stabilizing Automorphisms of a Finite Group presented by Mr. Bruce Staal has been accepted towards fulfillment of the requirements for Ph . D. degree inMathematics 7 7 7 r m ,, _ (JV/[égj/Cény ) Major professor Date December 13, 1974 / 0-7639 om ; ? "anemic wmwwmuu.; “moms ”nun-nu 3 ABSTRACT A STUDY OF THE STABILIZING AUTOMORPHISMS OF A FINITE GROUP BY Bruce Staal One of the most important objects of investigation in finite group theory is the group of automorphisms of a finite group. If one can say something about the automorphisms of a group then conceivably he can say something about the group itself. Our interest here is to consider certain relationships between subgroups and series of subgroups of a group and subgroups and series of subgroups of its automorphism group. Given a series G of subgroups of G; say age-=3 sz--- o 1 IV Gn = ( 1 ), we say that an automorphism a of G stabilizes G if a(x)x-l e G. 1+1 for all x 6 G1 for 1 = 0, 1, ° ° -, n - l. we obtain information about stabilizing automorphisms of a finite group building upon the work of Kaloujnine, P. Hall, and Baer. If A is a set of automorphisms of G then the multiplier group of A, M(G,A), is given by M(G,A) = G 1 2 o - -tz Gn = ( l ). Then if Ci is a n-group, a is a w-automorphism. 2. If a stabilizes a series of G and Fit(G), the Fitting subgroup of G, is a n-group then a is a n-automorphism. Using these results it is shown that the inner automorphisms that'stabilize some series of G are precisely those induced by elements of the Fitting subgroup. Kaloujnine defined the following generalization of a nilpotent group. For ASAut (G), let MO(G,A) = G and Mi+1(G,A) = M(Mi(G,A),A). If there exists a natural number n such that Mn(G,A) = ( l ) then G is said to be A-nilpotent. The usual idea of a nilpotent group occurs when A is the group of inner automorphisms of G. In Chapter III the stabilizing group, S(G), is defined to be the group generated by the stabilizing automorphisms of G. After examining some of the general properties of S(G) it is shown that S(G) is nilpotent if and only if G is S(G)-nilpotent. we then look at the conditions under which a group is Aut(G)-ni1potent and S(G)-nilpotent and conclude the chapter by considering the structure of S(G) for some of the well-known groups. A subgroup H of G is a Hall subgroup if the order of H and the index of H in G are relatively prime. Baer has shown that if Q: G = Ho 2 H1 2 ' ° ' 2 Hn = ( l ) is a series of Hall subgroups which Bruce Staal are normal in G, then an automorphism a of G stabilizes H if and only if a is an inner automorphism induced by an element of - n Z(H) =.H Z(H.). If G = H ° ° ° H with H.H. 1:1 1 l n l j H.H. and 3 1 1, ° ° °, n, the series (Inil. Injh = 1.1 n.1,;- -:= 000 on. 0.0 > = . H G Hl Hn 2 H1 Hn-l 3 H1 _ H6 ( 1 ) can be formed The series Baer considered have the above properties, but were also normal. we are able to get Baer's conclusion if H has a property which is an extension of the Sylow Tower property. It is also shown that Baer's conclusion holds for the inner automorphisms of G in the series H described above. The question of whether these inner automorphisms are the only stabilizing automorphisms for this type of series remains open. If G: G = Go 2 G1 2 ° ° ' Z Gn = ( l ) is a series of subgroups of G, an automorphism u of G is said to power stabilize G if for each x 6 G1 there is a natural number n and y e Gi+l such that a(x) = xny. This generalizes the idea of a stabilizing automorphism and it may be noted that if G is a subnormal series, a is a power stabilizing automorphism of G if and only if a induces a power automorphism of G1 GT. for i = 0, 1, ° ' °, n - l. The following results are obtained in 1+1 Chapter V in the direction of the results for stabilizing automorphisms. 1. If G is a subnormal series of subgroups of G then the set of power stabilizing automorphisms of G is supersolvable. 2. “If G is a normal series of subgroups of G then the multiplier group for G determined by the power stabilizing automorphisms of G is supersolvable. A STUDY OF THE STABILIZING AUTOMORPHISMS OF A FINITE GROUP BY Bruce Staal A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1975 I .~"} \/ ‘ 3 ACKNOWLEDGEMENTS The author wishes to express his appreciation to his major advisor, Professor w. E. Deskins, for his advice and guidance which made this thesis possible. He also wishes to thank Professor J. E. Adney for his many helpful suggestions. ii TABLE OF CONTENTS Page nnmmmnnm. ... ... ... ... ... ... ... ... .. 1 CHAPTER I: BACKGROUND, DEFINITIONS AND THEOREMS . . . . . . . . . 4 CHAPTER II: THE STABILIZING AUTOMORPHISMS OF A FINITE GROUP . . . 9 CHAPTER III: THE STABILIZING GROUP OF A FINITE GROUP . . . . . . 24 CHAPTER IV: AUTOMORPHISMS WHICH STABILIZE HALL-TYPE SERIES . . . 40 CHAPTER V: POWER STABILIZING AUTOMORPHISMS . . . . . . . . . . 4 50 INDEX OF NOTATION . . . . . . . . . . . . . . . . . . . . . . . . 63 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 iii INT RODUCT ION Most of our knowledge of abstract algebraic objects such as groups, rings, or vector spaces has been gained by studying either the substructures of the objects themselves or the homomorphic mappings into and out of the objects. The study of automorphisms of finite groups is an example of the latter of these two approaches. If one can say something about the automorphisms of a finite group then he can often say something about the group itself. In this paper we consider certain relationships between subgroups and series of subgroups of a group, G, and the elements and certain subgroups of the automorphism group of G. A well-known example of this kind of result is Theorem: If H is a normal subgroup of G and A is the set of automorphisms of G such that for each a e A the following hold: 1) a(h) = h for all h e H 2) a(g)g-l e H for all g e G then A is an abelian subgroup of the automorphism group. Here a normal subgroup of a group can be thought of as singling out a certain abelian subgroup of the automorphism group. If an automorphism, a, has the properties stated in the theorem with respect to a normal subgroup H, a is said to stabilize H in G. An automorphism of G is said to stabilize a subnormal series of G if it 2 leaves the series invariant and induces the identity automorphism on the factor groups. An equivalent definition can be given which is applicable to any (possibly non-subnormal) series. In Chapter I we will see that Kaloujnine, P. Hall and Baer have proved several results concerning stabilizing automorphisms and that as a result certain connections between the structure of the group and its automorphism group have been obtained. Here we set down the basic definitions and results which provide the background necessary for the remainder of the paper. In Chapter II those properties of stabilizing automorphisms which are possessed irrespective of the nature of the series stabilized are examined. we look at how restrictive the condition that an automorphism stabilize some series is. It is shown that a standard subnormal series of shortest length that is stabilized by a given stabilizing automorphism can be constructed. This aids in obtaining results about the order of a stabilizing automorphism in terms of the order of the group. Using these, a characterization of the inner automorphisms which are stabilizing automorphisms is found. These are precisely those inner automorphisms which are induced by elements of the Fitting subgroup of the group. In Chapter III we consider the kind of substructure the stabilizing automorphisms form in the automorphism group. To aid in this we define the stability group of a group G, S(G), to be the group generated by the stabilizing automorphisms of G. In the case when S(G) is nilpotent its structure is shown to have a very interesting relation to the structure of G. The structure of S(G) is examined for many of 3 the well-known groups and some relationships between the stabilizing group of G and those of its subgroups and factor groups are considered. In [ 2 ] Baer proved that only certain inner automorphisms stabilize a series of normal Hall subgroups. In Chapter IV we look at the kinds of automorphisms which stabilize series with some but not all of the properties of a normal Hall series. Some extensions of Baer's work especially for inner automorphisms are obtained. A stabilizing automorphism of a subnormal series induces the identity automorphism on factor groups. In Chapter V we consider automorphisms which induce power automorphisms on the factor groups of a subnormal series. It is shown that a group of such power stabilizing automorphisms for a subnormal series of a finite group is supersolvable. The multiplier group of a power stabilizing group of automorphisms for a normal series is also shown to be supersolvable. The notation used is mostly standard and an index of notation is included at the and of the paper. All theorems, lemmas, corollaries, l examples, and definitions are numbered consecutively in the order that they appear in each chapter with the first digit referring to the chapter number. The numbers in square brackets refer to the bibliography. CHAPTER I BACKGROUND DEFINITIONS AND THEOREMS Our study in this paper is restricted to finite groups; accordingly, G will denote a finite group throughout. We let Aut(G) denote the group of automorphisms of G and Inn(G) the group of inner automorphisms of G. If G is a group and H is a normal subgroup of G(H 9 G), then a e Aut(G) is said to stabilize H in G if u(h) = h for all h e H and u(g)g-1 e H for all g e G. It has long been known that the set of all such automorphisms for a given normal subgroup H form an abelian subgroup of Aut(G). An automorphism, a, with these properties is called a stabilizing automorphism of H in G and Stab(H,G) = { a:a stabilizes H in G } is called the stability group of H in G. In [ 8 ] Kaloujnine generalized this notion as follows. Let G: G = Go 2 G1 2 ° ° ° 2 Gn = ( l ) be a series of subgroups of G. We will say that G is a subnormal series of G (denoted G: G = G 2 G g - - - G = ( 1 )) if G. 2 G. o l n 1+1 1 for i = 0, 1, ° ° °, n - 1. An element a e Aut(G) leaves G invariant if O(Gi) = Gi' i = 0, l, ' ° -, n. If a leaves G invariant and G is a subnormal series then for each i = 0, ° ° - n - l, a e Aut(G) induces an automorphism, 5i, on the quotient group Ei_ defined by G1+1 ai(gGi+l) = a(g)Gi+1 for g e 61' Now if 0 leaves the subnormal series G invariant and induces the identity automorphism on Si_ for G1+1 i = 0, ° - °, n - l, we will say thattxstabilizes G and in this case call a a stabilizing automorphism of G. . . . . . G. . . Now a induCIng the Identity automorphism of _;. Is equivalent G1+1 5..= .=. .."r' to 1(XG1+1) u(x)G1+1 xG1+1 for all x 8 G1 his is, In turn, equivalent to c:z(x)x-1 e Gi+1 for all x e Gi' Using this equivalence it is possible to obtain a definition of stabilizing automorphism for any (possibly non-subnormal) series of G as follows. Definition 1.1: Let G be a group, G: G = G 2 G 2 ° ° ' Gn = ( 1 ) a series of G, and a e Aut(G). a is said to be a stabilizing_ automorphism 9£_G_if a leaves G invariant and for all x e Gi' e:(x)x'-l e Gi+l' i = O, 1, ° ° - , n - 1. Furthermore if there is a series of G which a e Aut(G) stabilizes, then we call a a stabilizing automorphism g£_§, The multiplier group is an important aid in the study of stabilizing automorphisms. Definition 1.2: If AS?Aut(G) then M(G,A) = O o o > > > ‘ ' ° . f. G YO(G) 2 “11(6) _ .. Yi(G) ., Yi+1(G) ,_ Is de med by Y°(G) = G and Yi+l(G) = [ yi(G),G ] which is the subgroup generated by all the commutators x-ly-1xy where x e Yi(G) and y c G. A finite group G is nilpgtent if there is an integer n such that Yn(G) = ( l ) and the least such integer is called the class of the nilpotent group. 7 In [ 8 ] Kaloujnine proves: Theorem 1.5 (Kaloujnine): Let G be a normal series of G of length m, then Stab(G) is nilpotent of class S m - 1. An example is given to show that this is the best possible estimate of the class of this nilpotent group. For the case in which G is not a normal series the best Kaloujnine could do was to show that Stab(G) was solvable of derived length 5 m - 1. Here we mean by the derived length of a finite solvable (n) group the least integer n for which the term G of the derived series is trivial: the derived series of G, G = 6(0) 3 G‘l) 3 - ° - z C(l).3 G<1+l) - ° - is defined by .+ . . G(°) = G and G(1 1) = [ C(l),G(l) ]. Of course, a finite group is solvable if there is an integer n such that G‘n) = ( 1 ). In [ 5 ] P. Hall was also able to improve this result with Theorem 1.6 (P. Hall): Let G be a series of G, then Stab (G) is nilpotent m(m-l) of class 5 , where G is of length m. 2 Hall also gives an example to show that this is the best possible estimate of the nilpotence class, i.e., there is a group with a non- normal series in which Kaloujnine's estimate of m - l for the nilpotence m(m—l) is needed. 2 class of the stabilizing group is too small and Thus it is apparent that the study of stabilizing automorphisms gives a connection between some of the nilpotent normal subgroups of G (M(G,Stab(G)) and some of the nilpotent subgroups of Aut(G) (Stab(G)). Definition 1.7: If n is a set of primes then a natural number n is called a n-number if its prime factorization contains only primes from n. A group G is a n-group if its order is a w-number and a e Aut(G) is a n-automorphism if its order is a n-number. The fOllowing result is known and it is a consequence of the stronger Corollary 2.16 in this paper. Theorem 1.8: If a s Aut(G) stabilizes a series of G and G is a n-group, then a is a n-automorphism. A subgroup H<=G is called a Hall subgrogp of G provided (IHI,[G:H]) = l where [G:H] is the index of H in G and [HI is the order of H. In [ 2 ] Baer proves the following concerning the automorphisms that stabilize a normal series of Hall subgroups of G. we use the notation Z(G) for the center of G and "x denotes the inner automorphism of G induced by x e G, i.e., nx(g) = x-lgx for g' e G. Theorem 1.9 (Baer): Let H: G = H6 2 H1 2 - ° - g Hn = ( l ) be a normal series of Hall subgroups of G. Then a e Aut(G) stabilizes H if _ n and only if a = n where x e Z(H) = H Z(H.). 1: i=1 1 Using these basic results as our starting point we shall obtain more information about the stabilizing automorphisms of a group. CHAPTER II THE STABILITY AUTOMORPHISMS OF A FINITE GROUP In this chapter we investigate the general properties of stabilizing automorphisms which are independent of a particular stabilizing series in order to discover what characterizes these automorphisms in the whole group of automorphisms. Suppose ASStab(G) where G: G= G02 G1 2 - ° ° 2 Gn= ( l) is a series of G. Now M(G,A); H M(G,a)ElGl. If we assume (15A M(Gi,a)€-.G. E = S S Mi(G,A) Gi, then Mi+l(G,A) M(Mi(G,A),A) M(Gi,A) II 1+1. aeA Thus it follows by induction that Mk(G,ATE Gk k = 1, ° - °, n. In particular Mn(G,A)E:Gn = ( l ) so Mh(G,A) = ( l ). Let m be the smallest natural number such that Mm(G,A) - ( 1 ). Let x e M(G,A), a e A and consider o:(x)x-'1 = y e M(G,A) then O(x) = y x e M(G,A) so for all a e A, a is an automorphism of M(G,A). Further, if z is a generator of M2(G,A), that is, z = c::(x)x-1 for some a e A and x e M(G,A) then both u(x) and x“1 are in M(G,A) so 2 e M(G,A). Hence M2 (G,A)§ M(G,A) . Proceeding inductively, we assume a induces an automorphism of Mk(G,A) for all a e A and Mk+1(G,A)E Mk(G,A). If x e Mk+1(G,A), then x e Mk(G,A) so that on(x)x-1 e Mk+1(G,A) for all u e A and hence a(x)e Mk+l(G'A) for all a e A. 10 If 2 is a generator of Mk+2(G,A), that is, z = u(x)x-1 for some a E A and x e Mk+1(G,A) then both a(x) and x.1 are in Mk+1(G,A) so 2 E Mk+l(G'A) and we have that Mk+2(G'A)£Mk+l (Gm. A series H can be constructed as follows: E:G=M(G.A)2M =M(G,A)B°°'2M =M(G,A)=(1) o l 1 m m This series is subnormal since M(G,A) 3 G and hence Mk+1(G,A) 3 Mk(G,A). The above argument then shows that A stabilizes H and Mk(G,A) is the smallest possible subgroup which can appear as the kth subgroup in a series which A stabilizes. This is summarized in: Theorem 2.1: Let ASStab (G) for a series G: G = G 2 G > ' ' ° 2 Gn = ( 1 ) of the group G. Then there is a least natural number m such that Mm(G,A) = ( 1 ) and the series F4:G=M 5M =M(G,A)1>---2M =M(G,A)=(1) o 1 1 m m is a subnormal series stabilized by A. Furthermore, H is a series of least length stabilized by A. From [ 1 ] we get the following: Theorem 2.2: If AGAuuG), then M(G,A) = M(G,). Proof: -1 -1 —1 If 0.8 e A, then aB(g)g = [ a(8(g))(8(g)) ] (B(g)g ) e M(G,A). Therefore M(G,)§ M(G,A). Since A S ,M(G,A)SM(G, ) so M(G,A) = M(G,). So we see that for many subsets of Stab(G) in Theorem 2.1 we may get the same series M. 11 It is natural to ask whether H.might indeed always be a normal series. The following theorem of P. Hall in [ 3 ] is used to show that this is not the case. Theorem 2.3 (P. Hall): There exists a nilpotent group G of class 2 with a series of subgroups of length 3 whose stability group is of class 3. Let B be the stability group of class 3 in the theorem. Form the series M: Mo = G 2 M1 = M1(G,B) 2 M2 = M2(G,B) 2 M3 = M3(G,B) = (1 ) Now M2 cannot be a normal subgroup of G for if it were then H would be a normal series and by Theorem 1.3 B would have to be nilpotent of class 5 2. An important special case of Theorem 2.1 is that in which A consists of a single stabilizing automorphism. We state this in Corollary 2.4: Let u e Aut(G) be a stabilizing automorphism of G. Then there is a least natural number m such that Mm(G,a) = ( 1 ) and the series M: MO=GPM1=M1(G,(1)B° ° 02Mm=Mm(G,a)= (l) is a subnormal series of shortest length which is stabilized by a. In what follows we will find a better estimate for the order of a stabilizing automorphism than that offrheorem 1.8 and characterize the stabilizing automorphisms of some well-known groups. 12 In the proof of the following theorem we will use the semi-direct product in a special case. we will give a restricted definition of such a product and the basic results we will use. A more general discussion of semi-direct products can be found in [ 10 ] pages 212-217. Let G be a group and A be an abelian subgroup of Aut(G). We form the semi-direct product K = G'IA] in the following manner. K is the set of all pairs (g,a) where g e G, a e A with binary operation (g,a)(h,B) = (gu(h),aB). Now K is a group and K has the property that if we form the inner automorphism of K induced by (1,u) (we will denote it by "a rather than n ) then this inner automorphism acts on (l,a) elements of G in K (i.e., (9,1)) in the same way as G acts on G as the following shows: (1,0,) (911) (110)-]- “(1(9! 1) (Mg) .00 (1.0?1) (0(9):1) The following theorem characterizes the stabilizing automorphisms of a p-group in terms of their orders: Theorem 2.5: If G is a p-group, then a is a stabilizing automorphism of G if and only if u is a p-automorphism. Pr_oo£: If a is a stabilizing automorphism the fact that a is a p-automorphism follows from Theorem 1.8. The converse is proved by showing that M(G,aY§ G when G is a p-group and a is a p-automorphism. Then since M(G,a) is a p-group and a is a p-automorphism of M(G,u) there must be a natural number m such that Mm(G,a) = ( l ). The series H is then stabilized by a proving the result. 13 Case 1; G is an abelian p-group M(G,a) =(a(g)g-1:g e G) = { ] = K. Since a is a p-automorphism, is a p-group so K is a p-group. Also since G is non-abelian so is K. Now the inner automorphism "a of K acts on the elements of G as a subgroup of K the same as G acts on the elements of G as a group. Thus M(G°[],na) = M(G,a) Now M(G- [<1>],ua)$ K' and 1059 [<1>]. K K I_ . _—____ . . . If K — G [<1>] then G'[<1>] K, but Since is CYclic and = GET§I37 it follows that ET-is cyclic. This is impossible since K K is non-abelian. So K' ¥ G' [<1>] and K'EG° [<1>]. Hence M(G° , "(1)5 K's G° [<1>] from which it follows that M(G,a)fi G. Theorem 2.6: If G is nilpotent and the factorization of G into the direct product of its Sylow subgroups is given by G = P1 x ' ° - x Pn where Pi is a pi-group, then a e Aut(G) stabilizes a series of G if and only if a P1 is a pi-automorphism i = 1, ° ° °, n. 14 Proof: cl Since P is an automorphism of P , a stabilizes a series of G i a if and only if Pi stabilizes a series of Pi for i = 1, ° ° -, n. By 1 Theorem 2.5 this is the case if and only if alPi is a pi-automorphism i = 1, ° ° °, n. Corollary 2.7: If G is nilpotent and g e G then «9 is a stabilizing automorphism of G. 2222.2: Referring to the factorization of G in Theorem 2.6 we write 9 e G as g = 9192 ° ° ' 9h, 91 e Pi' Since in a nilpotent group the Sylow subgroups centralize each other "9| = n = h pi 91 ' ° ' 9n|p1 which is a pi-automorphism and the result follows from the theorem. Thus the stabilizing automorphisms of a nilpotent group can be characterized in terms of the orders of the induced automorphisms on the Sylow subgroups. Example 2.8: This is an example to show that if G is not nilpotent the conclusion of Corollary 2.7 may not hold. Let G = $3, the symmetric group on three letters. Then Aut(SB) s S 2 Inn(S3). 3 Any stabilizing automorphism of 33 must stabilize S3 2 A3 2 ( 1 ) since this is the only non-trivial subnormal series of S3. But only the inner automorphisms induced by elements of A3 stabilize this series. Thus the remaining inner automorphisms are not stabilizing automorphisms. 15 It may be noted that this example also shows that the inner automorphisms need not be contained in the group generated by the stabilizing automorphisms. Eggmplg_g;23 This is an example to show that, in general, the set of stabilizing automorphisms is not a group. Let G = V the Klein 4-group. Only the automorphisms of order 2 stabilize a series of V. But Aut(V) = S3 and the elements of order 2 generate 83. A characterization is now given of the stabilizing automorphisms of any finite group using the characterization of the stabilizing automorphisms of a nilpotent group given in Theorem 2.6. Theorem 2.10: Let G be a finite group. a e Aut(G) is a stabilizing automorphism of G if and only if M(G,a) is nilpotent and alpi is a pi-automorphism i = l, - - -, k. Where Plx - . - ka = M(G,u) is the factorization of the nilpotent group M(G,a) into the direct product of its Sylow subgroups. moi: If a stabilizes a series of G then it follows from Theorem 1.4 that M(G,u) is nilpotent:by Corollary 2.4 a stabilizes 171: M°=G2M1=M(G,a) 2 - "'2Mn(G,a) = (1). Thus a stabilizes a series of M(G,a) and by Theorem 2.6 a P1 is a pi-automorphism i = l, - ° -, k. Conversely, since M(G,u) is nilpotent and a is an automorphism of M(G,a), according to Theorem 2.6 a stabilizes a series of M(G,a). Also a induces the identity automorphism on , so u stabilizes a _‘i__ M(G,a) series of G. 16 Note that Example 2.8 shows that the product of two stabilizing automorphisms need not be a stabilizing automorphism. The following shows when such a product is a stabilizing automorphism. Lemma 2.11: Let G be a finite group. If u,8 e Aut(G) are stabilizing automorphisms of G, then M(G,OB) is nilpotent. Pines By Theorem 2.10 M(G,a) and M(G,B) are nilpotent and it follows from Theorem 1.3 that M(G,a) and M(G,B) are normal subgroups of G. Let g e G. The identity -1 a8(9)9 1 = a(s(g)g'1)g’1> shows that M(G,aB)EM(G,a) M(G,B) so M(G,a8) is nilpotent. Theorem 2.12: Let G be a finite group. If a,B e Aut(G) are stabilizing automorphisms of G then OB stabilizes a series of G if and only if crap is a pi-automorphism i = 1, - - o, k where i M(G,aB) = Plx - - - ka, Pi the pi-Sylow subgroup of the nilpotent group M(G,aB). 2.1129 The Theorem follows from Theorem 2.10 and Lemma 2.11. The Klein 4-group shows that even for an abelian p-group one cannot get that a,B stabilize imply OB stabilizes for all 0,8 6 Aut(G). This indicates that restrictions must be placed on the stabilizing automorphisms themselves rather than on the group structure to get results in this direction. Theorem 2.13: If a,B e Aut(G) are stabilizing automorphisms of G and is nilpotent, then GB is a stabilizing automorphism of G. 17 2223: Since by Theorem 1.3 and Theorem 2.4 M(G,u) and M(G,B) are normal nilpotent subgroups of G, M(G,a) M(G,B) is a normal nilpotent subgroup of G. Let M(G,a) M(G,B) = Plx - - ° xPn be the factorization of H(G,a) M(G,B) into the direct product of its Sylow subgroups. a is an automorphism of M(G,a) M(G,B) since if x e M(G,a) M(G,B), c:t(x)x'-l e M(G,a) so a(x) s M(G,a) M(G,B). Similarly B is an automorphism of M(G,a) M(G,B). Since a and B stabilize the nilpotent group M(G,a) M(GpB): 8| 0 Pi and Pi are pi-automorphisms for i = 1, . 4 o, n. Let = le - ° - me be the factorization of into the direct product of its Sylow subgroups. Then write a = (a , ' ° ', a ) and 8 = (B . ° ' °. 8 ) where l m l m a. . e . i = 1 ° - - m. 1181 Q1! I I Since alPi and BIPi are pi-automorphisms there exists j such that “IP. = a. and 8IP. = 3,. So aBIP. = o‘IP. 8'P. = a.B. and i j i j i i i j j OBI aij 5 Q3. which is a pi-group. Thus Pi is a pi-automorphism for i = 1, ° ° ', n. Since M(G,GB)EM(G,O) M(G,B) the factorization of the nilpotent group M(G,a8) into the direct product of its Sylow subgroups can be written M(G,a8) = P1 x ° ° - xPr" where P'iS Pi for i = 1, - - o, n. 8I Then a . is a pi-automorphism for i = 1, - - 0, n. Hence, by Theorem P . l 2.12 GB is a stabilizing automorphism of G. 18 Corollary 2.14: If 0,8 5 Aut(G) are stabilizing automorphisms of G and d8 = Ba then GB is a stabilizing automorphism of G. me Since dB = Bu, is abelian and hence nilpotent. we now examine the orders of the stabilizing automorphisms and obtain results which extend Theorem 1.8. This will make it possible to obtain more information about automorphisms of a finite group which are stabilizing automorphisms. Theorem 2.15: Let G be a finite group and a e Aut(G). Suppose further that k is the least natural number such that Mk(G,a) = Mk+1(G,u) then a is a n-automorphism where 1r = n' U 1T" with ale(G,a) a 1r'-automorphism and M(G,u) a U"-group. Proof: Suppose H is given by M: Mo = G 2 M1 = M1(G,a) 2 - - - g Mk = Mk(G,u) = Mk+1(G,a) k being the least natural number such that Mk(G,A) = Mk+1(G,A). We proceed by induction on the length of M. Suppose first that M1(G,A) - MO(G,A). Then the theorem says that a is a n-automorphism a . . . . . where I is a n-automorphism which certainly is the case. M (GIG) = G 0 Suppose the theorem is true when H is of length k - 1. Consider a such that M: M0 = G 2 MllG,a) 2 ° ° ° 2 Mk(G,a) = Mk+1(G,a) is of length k. Now a is an automorphism of M1(G,u) and the H series for alMl(G,a) is of length k - 1. 19 By induction al M1(G,a) is a o-automorphism where O = n' U o" with ale(G,a) a n'-automorphism.and M2(G,a) a O"-group. If g e G, c:(g)g-1 = x e M1(G,a) so a(g) = xg and a(u(g)) = u(xg) = a(x)o(g) = a(x)xg. Then in general a2 (9) ll Q E an(g) ° ° ° u(x)xg. Let m be the least natural number such that um(x) = x. Then 01| «I since x e M1(G,a) a(x) = M1(G,a) (x) and, since M1(G,u) is a O-automorphism it follows that m is a O-number. Suppose M1(G,u) is a 1r"-group, then 0"E 1r". Now um+1(g) = x um-1(x) - ° ° a(x)xg and 2 mrl 2 a 1“(g) = (a (2:) ° ° ° a(x)X) g. tm mrl t So that, in general, a (g) = (a (x) - - - a(x)x) 9. Hence atm(g) = 9 if and only if (um.1(x) ° ° - a(x)x)t = 1. If m-1 . .. s = Id (x) ° ° - u(x)x| which is a W -number then sm is a O U 1r" = n'U O" U 1r" = 1r' U 1r" number since O"£1r". Set n = n'\’ n". Now let q be the n-number with the highest exponents appearing on the primes in the factorizations of the numbers ms Obtained as described above as g ranges over all of G. Then if g e G, m and s can be found as above and there is a a natural number a such that q = ams and uq(g) = (ems) (g) = 9. Since this is true for all g e G, '0' q. But q is a n-number and thus a is a fl-automorphism. This theorem has two corollaries which give information about the order of a stabilizing automorphism. 20 Corollary 2.16: Let G be a finite group and suppose a e Aut(G) stabilizes the series -: = > > . . . > = . ' ‘ G G Go _ Gl _ _ Gn ( l ) Then if G1 is a n-group, a is a nbautomorphism. 2:25.: Since a is a stabilizing automorphism of G there is a natural number n such that Mn(G,a) = ( l ). By Theorem 2.15 a is a n'-automorphism where M(G,o) is a 1r'-group. Now M(G,o)SGl so n'E 1t and hence a is a n-automorphism. It may be noted that when G is the particular series H the corollary says that if a is a stabilizing automorphism and M(G,o) = M1(G,u) is a n-group then a is a n-automorphism. The corollary which follows gives information about the order of a stabilizing automorphism,u, which is independent of the particular a chosen. Corollary 2.17: If a stabilizes a series of G and Fit(G), the Fitting subgroup of G, is a 1r-group then a is a rautomorphism. 2525a: a stabilizes H: G = M0 2 M1(G,a)2 - ° ° 2 Mn(G,u) = ( l ). By Theorems 1.3 and 1.4 M1(G,u) is nilpotent and Ml(G,O.) a G thus M1(G,a)§_ Fit (G). Then a stabilizes G 2 Fit(G) g M1(G,u)‘2,° ° - 2 Mn(G,a) = ( l ). So by Theorem 2.15 if Fit(G) is a negroup a is a nbautomorphism. Example 2.18: Let G = $3, the symmetric group on three letters. Now Fit(SB) = A3. Corollary 2.17 then says that any stabilizing automorphism of S is a 3-automorphism showing that none of the 3 2-automorphisms of S can stabilize a series of S . 3 3 21 We shall now determine exactly which inner automorphisms of a finite group stabilize a series of G. 1y-lxy denote the commutator of the elements Let [ x,y ] = x- x and y of G. We define inductively the following series of commutators in G. [x,y 10 = y, [x,y ]n = l x,[ x,y ln-l ]. we call an element x of G an Engel element of G if for each y e G there exists a natural number n(y) such that [ x,y ]n(y) = 1. Let Fit(G) denote the Fitting subgroup of G and define inductively Fit°(G) = ( l ), Fitn(G) is the subgroup of G corresponding to Fit(EEEfEL-—0. It's known that all elements of Fit(G) are Engel n-l‘G) elements and that if G is a finite solvable group, then Fit(G) is exactly the set of all Engel elements of G. Also Fitn(G) is a normal subgroup of G and solvable if G is any finite group. Theorem 2.19: Let G be a finite group. If for g e G, ng ,the inner automorphism induced by g, stabilizes a series of G, then 9 is an Engel element of G. Proof: Suppose g e G such that wg stabilizes a series of G. We must show that for each x e G there exists a natural number n(x) such that ( 91x 1 1° n(x) = Since ng is a stabilizing automorphism there exists a natural number n such that Mn(G,ng) = ( l ). Now [ g,x ] = g-lx-lgx = ng(x-1)x e M1(G,ng). Proceeding by induction su ose [ x ]. e M.(G n ) then PP 9' 3 3'9 22 [903‘ lj+1 = [go[ 91" Jj] -1 -l x a I o g [9. 13 9(9):]3 -1 1rg([g.xl jll 9.2: 13. thus [ g,x ]j+l e Mj+l(G,wg) therefore [ g,x ]n e Mn(G,ng) = ( 1 ). Since this is true for all x e G we conclude that g is an Engel element of G. Proposition 2.20: If for g e G, M(G,ng) is nilpotent then 9 e Fitn(G) where n is the least integer such that Fitn(G) = Fitn+1(G). Proof: If n is the least natural number such that Fith(G) = Fitn+1(G), then for x e G, ng(x-1)x e Fit(G) since M(G,ng) is nilpotent and normal. But Fit (G) E Fitn(G), so "96(1):: e Fitn(G) , that is -1 '1 G g x gx e Fitn(G) thus gFitn(G) e Z(Fitn(G))° __§_._c-__§._.= But Z(Fitn(G))'I'F1t(Fitn(G)) ( l ) therefore 9 e Fitn(G). The following theorem gives a characterization of the inner automorphisms which stabilize a series of G: Theorem 2.21: If g e G then Hg is a stabilizing automorphism of G if and only if g e Fit(G). Proof: If g s Fit(G), since Fit(G) s G, M(G,ng)!§ Fit(G). By Corollary 2.7, since Fit(G) is nilpotent, Hg stabilizes a series of Fit(G). Therefore ng stabilizes a series of G. 23 If fig stabilizes a series of G, we proceed by first considering solvable groups. Case 1; G is solvable. Suppose fig stabilizes a series of G. Then by Theorem 2.19, g is an Engel element and since G is solvable, Fit(G) is exactly the set of Engel elements of G. Thus 9 e Fit(G). I g§§g_3; G is any finite group. Let n be the least natural number such that Fitn(G) = Fitn+1(G). Since fig stabilizes a series of G, M(G,ng) is nilpotent. Then by Proposition 2.20 g e Fitn(G). Now Fitn(G) is solvable and Hg is an automorphism of Fitn(G), so since Hg stabilizes a series of G it must stabilize a series of Fitn(G). It then follows from Case 1 that g e Fit(Fitn(G)). Now Fit(Fitn(G)) is a characteristic subgroup of Fitn(G) which in turn is normal in G. Therefore Fit(Fitn(G)) S G and nilpotent so that Fit (Fitn(G)) E Fit(G) and g e Fit(G) . CHAPTER III THE STABILIZING GROUP OF A FINITE GROUP Example 2.9 shows that, in general, the set of all stabilizing automorphisms of a group is not itself a group. In this chapter the properties of the subgroup of Aut(G) generated by the stabilizing automorphisms will be investigated. By studying this group more information is obtained about stabilizing automorphisms and their relationship with the automorphism group. Definition 3.1: The stabilizing_group of the group G is given by < a a Aut(G): a is a stabilizing automorphism of G); This group will be denoted by S(G). Theorem 3.2: S(G) is a normal subgroup of Aut(G). Proof: If a is a generator of S(G), that is, a stabilizing automorphism of G, then there is a natural number n such that Mh(G,a) = ( l ). To prove that S(G) 9 Aut(G) it suffices to show that for each 8 e Aut(G) 8 ) = ( 1 ), since then u is B there is a natural number m such that Mm(G,o a stabilizing automorphism and therefore GB is in S(G). We use induction to show that B c -1 Mi(G,a )—B (Mi(G,a)) 24 25 Let g e G, then there is an h e G such that B 1 l B‘las(e’18'1(z’1) 8-1(u(z)z-1) Therefore Mi (G,u8) S B_1(Mi (G,u)) and Mn(G,aB)SB-1(Mn(G,a)) = ( 1 ) thus Mn(G,aB) = ( l ) completing the proof. From [ l ] we have the following: Theorem 3.3: If A is a normal subgroup of Aut(G) then M(G,A) is a characteristic subgroup of G. Proof: Let g e G, a e A, B s Aut(G). Then B. n Set E = { a , ° ° -, a }. By Theorem 2.2 M(G,E) = M(G,) = M(G,S(G)). Now M(G,E) = < ui(g)g-1: g e G, oi e E >. n So M(G,E) = II M(G,a.) i=1 1 Since a is a stabilizing automorphism by Theorem 1.4 M(G,ai) is nilpotent i = 1, ° ° °, n. Also M(G,ai) 4 G i = 1, - - o, n by Theorem 1.3. n Therefore H M(G,ai) = M(G,E) = M(G,S(G)) is nilpotent. i=1 Corollary 3.7: If S(G) contains a fixed point free automorphism, then G is nilpotent. 27 P222: If S(G) contains a fixed point free automorphism, then G = M(G,S(G)). Therefore by Theorem 3.6 G is nilpotent. The following example gives a simple application of Theorem 3.6. Example 3.8: Let G = 83, the symmetric group on 3 letters. Aut(SB) = S and if a e Aut(SB) has order 2 then M(S3,a) = S 3 3° Theorem 3.6 then gives that a k S(SB)° A generalization of nilpotent groups due to Kaloujnine in [ 8 ] will now be described. Definition 3.9: Let A S Aut(G). If there exists a natural number n such that Mn(G,A) = ( 1 ), then we will say that G is A-nilpotent. Further if n is the least natural number such that Mn(G,A) = ( l ), then G will be said to be A-nilpotent g£_ class a. It may be noted that in this generalization the usual idea of a nilpotent group occurs when A = Inn(G). Proposition 3.10: Let G be a finite group with H g Fit(G). Then G is A-nilpotent where A = { "h: h e H }. Proof: Since H S Fit(G) 9 G we have M(G,A) = [G,H] E [G,Fit(G)] E Fit(G) = 71 (Fit (G)). Proceeding by induction suppose Mj (G,A) E Yj (Fit (G) ) . 28 Then Mj+1(G,A) M(Mj(G,A),A) [Mj(G,A),H] 5 [Yj (Fit (G) ) ,H] S [Yj (Fit (G) ) .Fit (6)] Yj+l(Fit(G)) Thus Mk(G,A)S Yk(Fit(G)). Since Fit(G) is nilpotent there exists a natural number n such that Yn(Fit(G)) = ( 1 ). Therefore Mn(G,A) = ( 1 ) and G is A-nilpotent. The following theorem establishes that if S(G) is nilpotent, S(G) must stabilize some series of G. Thus if S(G) is nilpotent all_ of its elements are stabilizing automorphisms. In view of Definition 3.9 it may be noted that the following are equivalent properties of A S Aut(G). l. A stabilizes a series of G. 2. G is A-nilpotent. Theorem 3.11: If G is a finite group then S(G) is nilpotent if and only if G is S(G)-nilpotent. If G is S(G)-nilpotent, then S(G) stabilizes a series of G. Thus by Theorem 1.4 S(G) is nilpotent. Conversely, suppose S(G) is nilpotent. Form the semi-direct product H = G-[S(G)]. Let A = { "a e Aut(H): a e S(G) }. we now show that there exists a natural number n such that Mn(H,A) S G°[l] = G. 29 If we write elements of H in the form (g,a) where g e G and a e S(G) then multiplication in H of the second coordinates is independent of the first coordinates and is the multiplication of S(G). Now M(H,A) = [ H,( 1 )-S(G) ], so the second coordinate of M(H,A) is contained in Y2(S(G)). Proceeding by induction we assume that the second coordinate of Mk(H,A) is contained in yk+l(S(G)). Then Mk+1(H,A) = M(Mk(H,A),A) [ Mk(H'A)’( l )°S(G) ] The second coordinate of Mk(H,A) is contained in yk+1(S(G)) so the second coordinate of Mk+1(H,A) is contained in [ Yk+1(S(G)),S(G) ] which is Yk+2(S(G))- Therefore by induction the second coordinate of Mm(H,A) is contained in ym+1(S(G)) for m = 1,2, . . -. Then since S(G) is nilpotent there is an n such that Yn+1(S(G)) = 1 and hence Mn(H,A) S G- [1]. Let a be a generator of S(G), that is, a is a stabilizing automorphism of G. From the work above there is a natural number n such that Mn(H,fla)§E G'[l]. Also since a is a stabilizing automorphism of G there is a natural number m such that Mm(G,u) = ( l ). c o = Now Mn+j(H,1ra) 2 M(G,a) [1] so that Mm+n(H,1ra) ( 1 ). Therefore "a is an inner stabilizing automorphism of H. By Theorem 2.21 a e Fit(H) and then since a is any generator of S(G) we can conclude that ( 1 )'[S(G)] E Fit(H) . 30 It follows from Proposition 3.10 that H is A-nilpotent. A then stabilizes F4: H = MO(H,A) ; M1(H,A) g - - . g Mn(H,A) = ( 1 ). It will now be shown that Mi(G,S(G)) is contained in the first coordinate of Mi(H,A). Let a e S(G), x e G’so that n(x)):1 is a generator of M(G,S(G)). Now 1r0‘(x,1)(x,l)--1 e M(H,A) and 1ra(x,l)(x,l)-1 = c:(x,1). Proceeding by induction we assume Mk(Cpn,a) = Cpn-k 32 k Let x e Cpn-k' then x = amp for some natural number m then (1(ampk)a-mpk = ampk(p+l)a-mpk mpk+l = a mpk+1 Thus Mk+1(CPn,u) contains a for all m. k+l Therefore Mk+l(cpn'a) = z CPn-(k+1). In particular for k = n-l it follows that Mh_1(CPn.a) = Cpn-(n-1) = CD A ( l ). Exggple 3.14: This is an example to show that it may happen that M(G,o) S Fit(G) for all a e S(G) and yet S(G) does not consist entirely of stabilizing automorphisms. Let V be the Klein 4-group, then Aut(V) = S3 and Fit(V) = V. Then M(V,a) 5 Fit (V) for all a e Aut (V), while S(V) = 53 and is not all stabilizing automorphisms. The problem of when a group is Aut(G)-nilpotent will now be considered. Lemma 3.15: Let P be a p-group, then P is Aut(P)-nilpotent if and only if Aut(P) is a p-group. areas Suppose Aut(P) is a p-group. By Theorem 2.5 all automorphisms of P are stabilizing automorphisms so S(P) = Aut(P). Then S(P) is a p-group and thus is nilpotent so by Theorem 3.11 P is Aut(P)-nilpotent. Conversely, if Aut(P) contains a p'-automorphism it cannot stabilize a series of P by Theorem 2.5. 33 Lemma 3.15 may be restated as follows: a p-group P is Aut(P)-nilpotent if and only if it admdts no p'-automorphisms. Exagple 3.16: Let C be the cyclic group of order 2“. Now 2n Aut(Czn) = czxczn-l, so the lemma says C2n is Aut(Czn)-nilpotent. Examplg_3.l7: Let D be the dihedral group of order 8. Now Aut(D) = D, so again the lemma says that D is Aut(D)-nilpotent. Theorem 3.18: A finite group G is Aut(G)-nilpotent if and only if G is nilpotent and its Sylow subgroups are p-groups that admit no p'-automorphisms. 2223: Suppose G is Aut(G)-nilpotent. G must be nilpotent since otherwise there exists a g e G such that g k Fit(G) and by Theorem 2.21 fig could not be a stabilizing automorphism. Conversely, if G is nilpotent, let G = P x o - oxPn be the l factorization of G into the direct product of its Sylow subgroups. Then Aut(G) = Aut(Pl)x ° ° ° xAut(Pn). But G is then Aut(G)-nilpotent if and only if Pi is Aut(Pi)-ni1potent for i = 1, ° ° °, n. Lemma 3.15 then give the desired conclusion. we now consider when the group G is S(G)-nilpotent. Theorem 3.19: Let P be a finite p-group. P is S(P) nilpotent if and only if Aut(P) has a normal p-Sylow subgroup. Proof: Suppose Aut(P) has a normal p-Sylow subgroup SP. Then S(P) = Sp since S(P) is generated by all the p-elements of Aut(P). Therefore S(P) is nilpotent and by Theorem 3.11, P is S(P)-nilpotent. 34 Conversely, if P is S(P)-nilpotent, then S(P) must be a p-group. But S(P) includes all of the p-elements of Aut(P) and therefore it must be the unique p-Sylow subgroup of Aut(P). Example 2:395 This is an example of a non-abelian p-group P (p ¥ 2) which is S(P)—nilpotent. Let P be the non-abelian p-group of order p3 for p A 2 with generators a and b satisfying the following relations: 2 _ ap = 1, hp = l, and bab 1 = a1+p. Then according to [ ll ] (page 151), [Aut(Pfl' = p3(p-l). Thus Aut(P) has a normal p-Sylow subgroup and by Theorem 3.19, P is S(P)-nilpotent. we may think of Theorem 3.19 as saying that the problem of finding S(P)-nilpotent p-groups is the same as the problem of finding p-groups in which Aut(P) has a normal p-Sylow subgroup. This latter problem.has been solved in [ 6 ] in which it is shown that Aut(P) has a normal p-Sylow subgroup if and only if the invariants of P are distinct. Thus for P an abelian p-group we can state Theorem 3.19 as follows: Theorem 3.21: If P is a finite abelian p-group then P is S(P)-nilpotent if and only if the invariants of P are distinct. It is clear that this can be extended to any abelian group as follows: Theorem 3.22: If G is an abelian group, then G is S(G)-nilpotent if and only if the invariants of 8p are distinct for each p-Sylow subgroup, Sp, of G. 35 Some examples of S(G) for various groups will now be examined. The next theorem gives a large class of examples of S(G) and will give an example of a group G for which S(G) is not solvable. Theorem 3.23: Let P be an elementary abelian p-group of order p“, then S(P) = SL(n,p). 222:: According to Theorem 2.5, S(P) is generated by the set of p-automorphisms of P. GL(n, SL UMP) Now Aut(P) = GL(n,p) and l Thus since SL(n,p) g GL(n,p), SL(n,p) contains all of the p-elements of GL(n,p). So S(P) E SL(n,p). On the other hand SL(n,p) is generated by the transvections and the transvections of GL(n,p) all have order p. 50 SL(n,p) S. S(P). Hence SL(n,p) = S(P). Corollary 3.24: If P is an elementary abelian 2-group then S(P) = Aut(P). Proof: GL(n,2) SL(n,2) Therefore Aut(P) = GL(n,2) = SL(n,2) = S(P). Example 3.25: Let G be the elementary abelian 2-group of order 23. Then S(G) = SL(3,2) = A5. So for this group S(G) is not solvable. We now find S(Sn) where Sn is the symmetric group on n letters. For n z 5, Sn 2 An 2 ( l ) is the only sub-normal series of Sn and if x 8 Sn' fix is not the identity automorphism on An for any x e S so S(S ) = ( 1 ). n n 36 For n 2, Aut(SZ) = ( 1 ) so S(SZ) = ( l ). For n 3, only the automorphisms induced by elements of A 3 stabilize the series 83 2 A3 2 ( l ) so S(S3) = A 3. For n = 4, there are several possibilities for sub-normal series S using A , V, the Klein 4-group, and its subgroups. Checking 4 4 these it is found that S(S4) = V. The automorphisms induced on subgroups and factor groups by stabilizing automorphisms are now considered. Theorem 3.26: Let N S G, u e Aut(G) and 5 the automorphism of g-induced by a, then M.(§2&) = "n(G'“’N. “N T— Proof: we proceed by induction. Certainly, Suppose Mk(§fia) = Mk(G'a)N. N Then Mkfléfi) = < an?) 6:)“: I: e Mk(%,a) > and Mk+l(G'a)N = < a(x)x-1N: x e Mk(G,a) > if a(x)x-1N for x e Mk(G,a) is a N (G,a)N N generator of Mk+1 , then, by the induction hypothesis, 3: 5 ”S((g'a) . 37 Then there is a y E Mi(G,u) such that 1 l a (in (in " (y)y-1N 5 Mk+l(G'a)N. N i = ,7 an?) (if Therefore Mk+1(%35) = Mk+l(G'a)N. N Corollary 3.27: If a e S(G) and N 2 c; then a e 5(3). Proof : Since a + a is a homomorphism Aut(G) + Aut(g) it suffices to show that if a is a generator of S(G), then a s S(g). So let a be a stabilizing automorphism of G. Then there is an n such that Mn(G,a)N = ( l )N = M (G,a) = ( 1 ) and by Theorem 3.26 M (9:5) = ( l ). n n N N N Therefore a e S(g). If we consider the homomorphism ¢: Aut(G) + Aut(g) defined by - G ¢(a) - a, the corollary Shows that ¢|S(G): S(G) + S(fi). Now Keri» = < a e Aut(G): M(G,d)$ N > so Ker¢|S(G) = < a e S(G): M(G,d)EN >. S (G) . Ker¢|S(G) Thus 8(3) contains a subgroup isomorphic with Theorem 3.28: If H S G, a e Aut(G) with a H an automorphism of H, then a eS(G)implies alH e S(H). Proof: C. MimmlH) .. Minna) . 38 Since a e S(G), there is an n such that Mn(G,a) = ( 1 ), thus Mn(H,a|H) = (1) and GIH e S(H). Suppose H is a characteristic subgroup of G, then 9: Aut(G) + Aut(H) defined by ¢(u) = OIH is an homomorphism. In particular, the theorem says that TI is an homomorphism of S(G) S(G) S(G) + S(H). So S(H) contains a subgroup isomorphic with ----- Ker¢|S(G) where Ker¢| = < a 6 8(6): aIH = l >. S(G) Definition 3.29: A group is characteristically_simple if it contains no proper characteristic subgroups. Theorem 3.30: If G is a characteristically simple finite group then either S(G) = ( l ) or G is an abelian p-group. £1299 By Corollary 3.4 M(G,S(G)) is a characteristic subgroup of G. So if G is characteristically simple M(G,S(G)) = ( l ) or G. If M(G:S(G)) ( 1 ), S(G) = ( 1 ). If M(G,S(G)) G, by Theorem 3.6 G is nilpotent. Since the p-Sylow subgroups of a nilpotent group are characteristic, G must be a p-group, but then Z(G) is non-trivial and characteristic, so G is abelian. From C 10] we have that a finite abelian p-group is characteristically simple if and only if P is not cyclic and of order pn n > 1. H Theorem 3.21: If g-is a characteristic factor of G, then 5(2) ( 1 ) or S(gb = SL(n,p). 39 Proof: Since gis a characteristic factor, it is characteristically simple and the direct product of isomorphic simple groups. By Theorem 3.30 80%) = ( l ) or gis an abelian p-group. But the only abelian p-groups that are direct products of isomorphic simple groups are elementary abelian p-groups. Then by Theorem 3.23 S(%) = SL(n,p) where I%I = pn. CHAPTER IV AUTOMORPHISMS WHICH STABILIZE HALL-TYPE SERIES It was noted in Chapter I that Baer has shown in I 2 .7 that if H:G=H 2H 3 . . . g H = ( l ) is a series of normal Hall 0 l n subgroups of G, then a e Aut(G) stabilizes H if and only if a = "x n where x e Z(H) = H Z(Hi)° In this Chapter we seek to generalize this i=1 result. We will examine the stabilizing automorphisms of series of the -o = 000 > coo >00.) > = form H. G H1 Hn - H1 Hn-l , _ H1 _ Ho ( l ) where HiHj = njai and (Ini|,|ajl) = 1 i ,4 j i,j = 1, - - -, n. We may note that the series which Baer examined do enjoy these properties but were also normal. We now describe how an interesting special case of this generalization may be obtained from Hall's work on solvable groups. Definition 4.1: A collection of Sylow subgroups { P ° ° °, Pn } of G 1' is called Sylow basis for G if the collection is pairwise permutable (Pin = PjPi' for i h j i,j = 1, ' ° °, n) and Hall has shown that G is a solvable group if and only if it has a Sylow basis. Hall showed further that any set of permutable Sylow 4O 41 subgroups of a solvable group G is included in some Sylow basis of G and two Sylow bases for a given solvable group are conjugate. Given a Sylow basis { P - - -, Pn } for a solvable group, 1' series of the following type can be constructed P: G=P1~ ooanzpl P g . . . > P 3 P = ( l ). These n-l 0 series have the properties of the series we will consider in this chapter. We recall the following definitions to be used in what follows: Definition 4.2: A finite group G is said to be a Sylow Tower Group if there is a normal series G: G = Go 2 G1 2 - - - g Gn = ( l ) with the property that for i = 1, - ° -, n, Gi is isomorphic with a Sylow subgroup of G. G i+l All Sylow Tower Groups are solvable and the class of all such Groups properly contains the supersolvable groups. Definition 4.3: For H 5 G, we define the core of H in G, written Core H, as Core H = '1 Hx. G G xeG This says that Core H is the largest subgroup of H which is G normal in G. we now prove some results in the direction of those of Baer. The following theorem will be used to obtain Baer's result for Sylow Tower Groups. 42 Theorem ii: Suppose G= H1 0 - - Hn' (IHiI'IHjI) = 1 i A j and HiHj = HjHi, i,j = l, ' ° °, n. Furthermore suppose there exists an ordering of the Hi such that a.: G = H o o o H. 2 H. o o o H g o o o 2 Hi 2 ( 1 ) l n 1 n-l 1 is a normal Hall series of G. Then if a e Aut(G) stabilizes H: G = H o o o H 2 H1 0 o o Hn—l 2 o o o 2 H1 2 ( 1 ,' a = 1r where x e Z(H) = ,II Z(H.). x i=1 1 Proof: we will show that a stabilizes Hi and use Baer's result in Theorem 1.9. Since normal Hall subgroups are characteristic n(H. o o o H. ) = H O O o H j = 1' o o o, n. from above u(xi - - - x. ) e H ° ° ° H. . 1 j-l 1 j-l Also a(x. )XTl e H. ° ° ° H . i. i. . J J l J 43 Thena(x.)lee(H ---u. flu ---H )SH °°°.H i. 1 1 '. n(x. ---x.)(X. ---x) eH oooH, J. 1. 1 for j = 2, ° ° °, n. O O O c O O O M(Hl Hil,a)._ H1 H111 and E o o o c o o 0 mai ,a) M(H1 Hi .00... H H l 1 1 Since H. is characteristic a(H, ) = H. and 11 11 11 NH. ,a) E H. . 1 1 1 1 Therefore M(H. ,a)E-H. n H - ° - H. = (1 ) so that 1 1 1 1-1 1 1 1 a is the identity on Hi . 1 Hence a stabilizes E1 and by Theorem 1.9 a = fix _ n n - x 6 Z(H.) = H Z(H. ) = H Z(H.) = Z(H). Corollary 4.5: Let G be a finite Sylow Tower Group and P1, - ° °, Pn be a Sylow basis for G. If a stabilizes P:G=P°'°PzP°°°P )ooo)P2(1)'then 1 n 1 n-l ' ‘ 1 _ n a = n for some x e Z(P) = H Z(P.). x . i 1=l Proof: A Sylow Tower Group has the property that its Sylow bases satisfy the hypotheses of Theorem 4.4. 44 Proposition 4.6: Let G be a finite group and H < G, a e Aut(G). Then if M(G,a) E H, aIH is an automorphism of H. Proof: Let h e H. o:(h)h-1 e M(G,aJSE H so n(h) e H. The following result shows that a non-trivial central automorphism cannot stabilize series of the type now being considered. ll :1: Theorem.2;15 Suppose G 1 0 . - H (IHiI'IHjI) = 1 n! - - -, n. If a e Aut(G) is a II :1: :1: p. C. II 3‘ 1* Jr and Hill). central automorphism and a stabilizes i: G = H - - - an > - . . > H > 1, then a = ( 1 ). 1 1 Proof: Z(G) 9 G so for i = 1, ° - 0, n Z(G)Hi is a group. By Proposition 4.6 a is an automorphism of Z(G)Hi. Now (I: Z(G)Hi:I-Ii], IHiI) = 1 so Hi is complemented in Z(G)Hi. Thus there exists 2 e Z(G) such that a(Hi) = Hi = Hi' Therefore C2 But since a stabilizes E C o o g c o o o c O O O = = . = O O O M(Hi,a)._ H Hi-l" Hi ( l ) thus a l on Hi 1 1, , n. 1 The following characterizes those inner automorphisms that: stabilize E. 45 Theorem 4.8: Suppose G = H 1 HiHj = HjHi 1,3 = 1, ' ' ', n and H: G = H1 ° ° ° Hn 2 H1 ° ° ° Hn-l Then for x s G, “x 1 ° ° ° xn, xi 6 Hi with xj e Z(H1 - - - Hj) and x1 0 o 0 fl 6 coreH .(Ijl.ono 0 HR) 3 = l, o o o, n l k+1 k = 1' o o 0' n-1. (*) Proof: The proof is by induction on the length of H. For n = l, "x stabilizes l < Hl if and only if x e Z(Hl)' that is, "x = 1. Let x = x ° ° ° x satisfy (*) then u = n 1 n x x ° ° ° x 1 n-l Since xn e 2011 ° - - Hn) = Z(G). So nxl . . . xn_1 = fix is an automorphism of H1 ~ ° ° Hn-l' By the induction hypothesis, since x1 5 Z(H1 4 ~ - Hi» i = 1, - - -, n-l. xl - - - xj c CoreH .(Hlo' ° - Hj) 3 = l, 1 j+1 ' ' o o o > o o o > > stabilizes H1 Hn-l _ _ Hl _ It remains to show that M(G,1rx)S H 1 ° ° ° Hnr (IHiIIIHjI) ( 1 ). =1 i‘jl > > . _Hl_(1) stabilizes H if and only if 1 Let g E G, wx(g)g- = X- 9 x 9- = (x o o o x )-1 (x X ) - 1 9 1 n g = (x x )- (x x ) -1 1 . . n-l g 1 O . n-l 9 since xn e Z(G). But (x . . x ) -1 H . H since 9 1 n-l g E 1 n-l x1 . . . xn_1 6 Core (H1 . . . Hn-1)° H o o o H 1 n -1 o o o c o o o . So nx(g)g 5 H1 H and M(Gmx) __ H H n-l 1 n-l Conversely, suppose "x stabilizes H. Then "x also stabilizes H . . . H > . . . > H > ( 1 )- 1 n-l 1 So on H . . . H n = n where l n 1 x y1 . . .Yn-l y e Z(H1 . . . Hi) i = 1, . . ., n-1 and y . . . y. 6 Core (H . . . H.) j = l, . - -, n-l by the induction 1 3 Heel.“ 3 1 j+l hypothesis. Hence n -1 = 1 _ x(Y1 ' ° ' Yn-1) H1 ' ' ° Hn-1 ‘ n -1 on H . . . H .0. x - (yl yn_1) 1 n l thenn -1(Y---Y)= (y1 . . . yn_1) x 1 n 1 x'1( )( )( )‘1x y1 ’ ° ' yn-l y1 ' ' Yn-l y1 yn-l = Y1 ' ' ' Yn-l' 50 W1 ' ‘ ‘ y ). (**) n-l “‘1 Write x = x ° ° ° x , x. 6 H. l n i 1 then " . . , -l(x ' ’ ° x _ ) = x ' ° ° x _ . x(y1 yn_1) 1 n 1 1 n 1 So ( o o o )(x o o o x )-l(x o o o x )(x o o o x) Y1 yn-1 1 n 1 n-l 1 n ( o o o )_1 — o o o x y1 yn-l - x1 n-l md( o o o )x-1(x o o ox)( o o o )-1=x o o ox y1 Yn-l n 1 y1 Y 1 n-l' Hence (x o o o x )( o o o )-l — x( o o o )-1(x o o o x ) 1 n y1 yn-l n Y1 yn-l 1 n-l ’ BY (**) we get [(Y' ° ° ° y )'l(x ° ° ° x )‘]x = 1 n-l 1 n-1 n x[(y---y )'1(x---x )1 (***) n 1 n-l 1 n-l Therefore n -1 = E (Y1 yn_1) (x1 xn_1) ]x n -1 x“Y1 yn-l) ("1 xn-l) 3 Now If H1 ° ° ° Hn-l is a fl-group n( . . . )-l(x . . . x ) is a n-automorphism. Y1 yn-l 1 n-l . O O O = . '- But Since (IHnI,IH1 Hn—ll) 1, Hn is a n group so that n is a n'-automorphism and (In |,| . . _ -1 . . . I)= l. xn xn le Yn-l) (x1 xn_l) Now '1! | In 0 O O -1 O O 0 Ion xn (Y1 Yn-l) (x1 xn-l)xn O O O H . H1 n-l By Corollary 2.16 Since "(y . . . y )-l(x . . . x ) stabilizes 1 n l n o o o > o o o > o o o > ) H1 Hn—l ’ H1 Hn-2 ‘ ‘ H1" ( l ) In . . . -l . . . I has the same primes as (Yl Yn) (x1 xn) 0 o o ' .. ' o o o H . . IH1 Hn-ZI and so is a n automorphism on H1 n-l Since fix is a n'-automorphism of H1 - - - Hn-l it must then be the identity on n H ° - . H , that is, x e CG(Hl - - - H ). l n-1 n-1 Now flx(x1 ° ° ° xn-l) = (x1 ° ° ° xn-l) so xn(x1 . O . xn—l) = (x1 . . . xn-1)xn and fix "x . . . x = "x . . . n so that Inx I [fix]. n 1 n-l 1 xn-l xn n Again by Corollary 2.16 since fix stabilizes to. )H000 >000) ) ' — ' Hl Hn - 1 Hn—Z _ _ H1 _ ( 1 ), fix is a n automorphism of G. So since fix is a n'-automorphism it must be that w = l on H ° ° ° H = G. x l n n Therefore "x . 0 . x = “X . . . x 1 n l n-1 and fix . . . x stabilizes H and by the induction hypothesis 1 n-l xi 8 Z(Hl . . . Hi), i = 1' . . .' n‘l. x ' ' ' x. E Core (H ' ° ° H.) j = l, ' ° ', n-2. J H J 1 j+l We already have that xn e Z(H1 ° ° - Hn). So we need only show that x1 ° ' ° xn-l 6 Core (H ° ° ' H ). This says that all the conjugates of x ° ° ° xn_ are in H1 ° ° - Hn-l so that x1 - ° ° xn_1 6 Core (H1 - ° ° Hn-l” Whether these are the only automorphisms that stabilize this type of series is an open question. CHAPTER V POWER STABILIZING AUTOMORPHISMS Let G: G = Go 2 G1 ° ° ° 2 Gn = ( 1 ) be a series of subgroups of G. Definition 5.1: a e Aut(G) is a power stabilizing automorphism of a if for each x 6 G1 there is a natural number n and y e G1+1 such that n(x) = xny. In case G is a subnormal series, Definition 5.1 can be restated as: o e Aut(G) is a power stabilizing automorphism of a if a induces a power automorphism of Gi for i = O,l,° ° -, n-l. G1+1 Theorem 5.2: If a and 8 are power stabilizing automorphisms of a, then so is 38. Proof: _ n Let x 6 G1 aB(x) — a(x y). Y E Gi+1 = o(xn) a(y) n m = (x) yaw). y' e Gi+1 = x““‘(y'a(y)) Now y'a(y) e Gi+ so 08 is a power stabilizing automorphism of a. 1 Theorem 5.2 makes possible the following: Definition 5.3: The power stabilizing group of a, denoted by PStab(E), is PStab(§) = { o:a is a power stabilizing automorphism of C }. 50 51 we proceed to work out the basic properties of PStab(G). Let PAut(G) denote the set of power automorphisms of G which is a normal subgroup of Aut(G). The properties of power automorphisms can be found in [ 4 ]. Theorem §;g; If G is a series of characteristic subgroups of G, then FStab(§) s Aut(G). .Pr_oo£= Let a e PStab(G), e e Aut(G) Then a induces Bi a power automorphism of Gi . Let 51 be the G1+1 automorphism of G1 induced by 6 (since Gi and G1+1 are characteristic G1+1 subgroups of G). -1 '31—"' . . G. . Then 6 aei = 61 oiei is a power automorphism of 1 Since G i+1 PAut(Gi ) g Aut(Gi ). Gi+l G1+1 So 6-106 is~a power stabilizing automorphism of G. §§23212_5.5: Let G = S and G: 83 2 A3 a ( l ). Then every automorphism -——- 3 of $3 is a power stabilizing automorphism of G, that is, PStab(G) 3 8 Thus PStab(G) need not be nilpotent and Theorem 3. 1.6 does not extend for PStab(G). Theorem 5.6: If a e Aut(G) is a power stabilizing automorphism of GI : G = G0 a G1 2 ° - - z Gn = ( l ) and G is a refinement of G, then a is a power stabilizing automorphism of G. 52 are-1: Let H S K.be two successive groups in 3. Then since a is a refinement of G there is an i such that G H < x g_G Let k e x. 1+1 5 - 1' Since k e 61' n(k) = kny, y 2 61+ So n(x) = K and since 1. Gi+1S H, y e H. Therefore a is a power stabilizing automorphism of G. Theorem 5.7: Let G: G = Go 2 G1 3 - - . 3 Gn = ( 1 ) be any series of G. If G: G = Go 2 G1 2 - - . > G = ( l ) is a series of G with G1 some of the groups in G, then Stab(G) g PStab(G). j are-2: Since G is a refinement of 6, Theorem 5.6 gives that PStab (E) s PStab (a) . Let 6 e PStab(§), a e Stab(é). It suffices to show that 97%6(x) = xk, k e G. for all x e G. . i. 1. 3+1 3 Let x e G. 6-1a6(x) 6-1a(xmy) y e G. 1 O 1 O 3 3+1 -1 6 (imyz) z 6 G1 j+l -1 _ e (me)6 1(z) -1 x6 (2) e'1(z) a G1 1+1 Therefore Stab(G) 9_PStab(G). Corollary 5.8: If G: G = Go a G1 3 . . . > G = ( l ) is any series of G, then Stab(é) g PStab(E). Proof: This is Theorem 5.7 with E = G. 53 We now describe briefly how Baer's work on supersolvable imersion in l 3 J may be used to prove that PStab(G) is supersolvable if G is supersolvable. Definition 5.9 (Baer): A normal subgroup H of G is supersolvably immersed in G if for every homomorphism 6 of G with He * ( 1 ) there is a cyclic normal subgroup A 1 ( 1 ) of G6 such that Theorem 5.10: If H 3 G and H has a series H: H = H 2 H 2 - o - 3 Hn = ( 1 ) with Hi cyclic of prime H1+1 order i = 0, ° ° -, n-1 and Hi 2 G i = 0,1, - - o, n then H is supersolvably immersed in G. Proof: Let 6 be a homomorphism of G with He ¥ ( 1 ). Then there is a largest i such that Hie H ( l ). Now H: is cyclic and Hi6 3 Ge, so H is supersolvably immersed in G. The following Theorem is from [ 3 ]: Theorem 5.11 (Baer): If H is supersolvably immersed in G, then the elements of G induce in H a supersolvable group of automorphisms. Let G be supersolvable and G: G = G 2 G P - - - e Gn = ( l ) be a chief series of G. Then Gi G1+1 is cyclic of prime order. Form the semi-direct product G-[PStab(G)], then using Theorem 5.10 we get that G is supersolvably immersed in 54 G'[PStab(G)J. It then follows from Theorem 5.11 that the elements of G-[PStab(G)] induce in G a supersolvable group of automorphisms. Since this group contains a subgroup isomorphic with PStab(G), PStab(G) is supersolvable. This conclusion is stated as Theorem.§;1g; Let G be a supersolvable group and G a chief series of G, then PStab(G) is supersolvable. Corollary 5.13: If G is supersolvable and G is any series of G, then PStab(G) is supersolvable. Proof : Refine G to E a chief series of G. By Theorem.5.6, PStab(G)§E PStab(§) and by Theorem 5.12 PStab(é) is supersolvable. Therefore PStab(G) is supersolvable. we will now show that if G is solvable and G is any subnormal series of G, then PStab(G) is supersolvable. Because of the length of the argument, it is convenient to split up the proof. We proceed by proving the result for composition series (Lemma 5.15 and Theorem 5.16) and then appeal to Theorem 5.6 to obtain the final conclusion (Corollary 5.17). EEEEB.§;l23 If G is a non-abelian simple group, the PAut(G) = 4 1 ). £225.: Let a e PAut(G). Then since power automorphisms are central, if g e G, o:(g)g'-1 e Z(G) = ( l ) because G is non-abelian and simple. Therefore a is the identity automorphism. Lemma5.15: LetE:G=G 2c; 9000 = b .t. o 1 - B Gn ( 1 ) e a comPOSi ion PStab (5) Stab(G) is supersolvable. series for G. Then 55 Proof: Define a series of subgroups of PStab(G) as follows: cn = PStab(§) C1 = { a e Ci+1: o induces the identity on 61 } i = 0,1, ° ° -, n-l. G1+1 II O ‘ O O O Q :3 Then co = Stab(G). We now show that ci a PStab(G) i Let a e Cn-l' B e PStab(G), x e Gn-l' Then B-la8(x) B-la(xp), B is a power automorphism of Gn- 1 -1 n . n B (x ),ofixes x e Gn-l thus cn_ g PStab(§). 1 Proceeding by induction suppose that C1+1 3,PStab(G). Let a 6 Ci' 8 e PStab(a)p X e Gi B-lo(xny) y e Gi+l' B is a power automorphism of Gi Gi+1 B'la3(x) B-l(anY') Y' e G. , a fixes Gi element-wise 1+1 6—— 1+1 -1 I ... - '1 I x3 (y ) 8(Gi+1) - G1+1 implies 8 (y ) e Gi+1° So 8-108 induces the identity automorphism on Gi . Since we G1+1 . It then follows are assuming C 1 for C. a e i+ - -1 g_PStab(G), 8 a8 a Ci+1 1+1 that ci 3 PStab(§). C. . . . . We now show that i is isomorphic with a group of power C1+1 automorphisms of Gi-l. Gi 56 Define y‘aCi-l) to be the automorphism of Gi-l induced by a for G i o e C.. i , then aB-l e C. so aB-l induces the identity If oCi_ = BC 1_1 1 i-l automorphism on Gi-l. G: Therefore a and B induce the same automorphism of Gi-l. V is Gi then a well-defined mapping and a homomorphism of Ci into Gi-l. Ci+1 Gi Suppose aCi_ and BCi_ are mapped to the same automorphism of 1 1 i-l. Then if x e Gi- and a and E are the induced automorphisms of Gi 1 a and B respectively on Gi-l G i 6(xci) = E=G d =( ).: 000 ... Then xn—l n-l an Gj Gj+1'xj 3 0,1, , n 1 For J = k: ‘ ° '1 n 0(Xj) = xj Since a e Sk. 59 Since a 5 U1 a(xk_1) = xk_1 gi, some 91 6 G1 so write n gi - xi gi+1, some g1+1 e Gi+l' n an integer then ( ) = n “ “h-1 xk-i xi gi+1° Choose a so that In] is minimal. Let B 8 Ti' Then as with a above = m I ' I 8(xk_1) xk_1 xi gi+1, m an integer, g1+1 e Gi+l° If r 2 (m,n), there exist integers s,t such that r = mt + ns. Then a38t(xk_1) = osst'1(xk_1 xim 9511) = ath-2(xk_1 xim 9i+l xgn gi+1) = asst'zhs.-. x3” 91...» 91.. . Gi+1 s tm a (xk_1 xi k), k e G1+1 tm+sn ‘ xk-1 xi h' h e G1+1 = xk-l “it h By the minimality of In] and since |r| < Inl it must be that [r] = In]. Then nlm and there is an integer a such that m = na. Since B("k—1) = xk-l “in 91+1 - - -1 xk_1 = s 13(xk_1) a 1 i i+1 -1 m , . . . . B (xk-l) xi gi+1, Since 8 is the identity of Gi 60 B( >— (““f1 3° xk-l " xk-1 “1 91+1 xk-l g1+1 xi ‘1“ xk-l xi k' k E Gi+l _ ) = oa(xk_1 3;“ k) m-m xk-l xi xi h' h e G1+1 = xk-l h a -l = Therefore a B e Ti-l so 8 Ti-l (a Ti- ) and Ti is cyclic, completing the proof. Ti-l Corollary 5.17: Let G be solvable and G be a series of subnormal subgroups of G. Then PStab(G) is supersolvable. m: Refine G to G a composition series for G. By Theorem 5.16, PStab(G) is supersolvable. Then by Theorem 5.6 PStab(G)§E PStab(G). Thus PStab(G) is supersolvable. Subsequent to the proof of Corollary 5.17 Schmid proved the following Theorem in [ 9 ] which makes it possible to generalize Corollary 5.17 for any finite group. Theorem 5.18 (Schmid): An automorphism group of a finite group G which normalizes (fixes) a composition series of G and induces a supersolvable group of automorphisms on the non-abelian composition factors is supersolvable. 61 The following proof was suggested by Professor Deskins: Theorem 5.19: Let G: G = G 2 G > ' ' ' 2 GD = ( l ) be a composition series for G, then PStab(G) is supersolvable. Proof: G. - . - . . Let i be a non-abelian factor of G. Since G is a composition G i+l . G . . G. series for G, i 13 Simple so Z( i ) = ( l ). Gi+1 Gi+1 - . G. Now a e PStab(G) induces a power automorphism on i and Cooper Gi+1 . has shown in [ 4 ] that a power automorphism is central. But G O O C 0 Z( i ) = ( 1 ) so a induces the triVial automorphism on Gi . Therefore G. G. i+l i+1 by Theorem 5.18 PStab(G) is supersolvable. Corollary 5.20: If G is a subnormal series of G then PStab(G) is supersolvable. greet: Refine G to G a composition series for G. Then by Theorem 5.19 PStab(G) is supersolvable and by Theorem 5.6 PStab(G) E PStab(G) so PStab(G) is supersolvable. If G is a normal series we can determine the structure of M(G,A) where A E PStab(G). Theorem 5.21: If G: G = Go 2 G1 2 ° ° ° 2 Gn = ( l ) is a normal series of G and o e PStab(G), then the inner automorphisms induced by oz(g)g-l are power stabilizing automorphisms for all g e G. 62 Proof: K G. Let g 6 G and G be a subgroup of _1_ . '+ 1 1 G1+1 9 Now'(g}-9gGi+l = K is also a subgroup of Gi since 1+1 1+1 4 a, G. 1 - G. O - O G. Now a induces oi a power automorphism of i so Gi+1 G 3i K 9 1+1 Kg K9 1+1 1+1 1+1 -1 gas x Therefore G—— = G . 1+1 1+1 Hence fla(g)g-1 induces a power automorphism of G1 and G1+1 "o(g)g-1 is then a power stabilizing automorphism of G. Theorem 5.22: Let G: G = GOP G1 2 - ° - 2 Gn = ( 1 ) be a normal series for G. If AE PStab(G) , then M(G,A) is supersolvable. Proof: By Theorem 5.21 if B = < -1: a e A, g e G>, 1101(g)g B S PStab (G). Then B is supersolvable by Theorem 5.19. z M(G,A) “W B M(G,A)n Z(G) ' M(G,A) M(G,A)fl Z(G) So is supersolvable. From M(G,A) n Z(G) S Z(G) it follows that M(G,A) is supersolvable. 1. Relations: 92 c: .~. IA (A ‘A E II. Operations: G“ or o(G) 90 or n(g) E H X [GzH] (m.n) INDEX OF NOTATION Is a subset of IS a proper subset of Is a subgroup of or is less than or equal to Is a proper subgroup of Is a normal subgroup of Is an element of Image of the set G under the mapping a Image of the element 9 under the mapping a Factor group Direct product of groups Index of H in G Subgroup generated by Set whose members are The number of elements in G The order of the element 9 The greatest common divisor of the integers m and n Restriction of the mapping a to the set H 63 64 III. Groups and Sets: Aut(G) The automorphism group of G Z(G) The center of G Fit(G) The Fitting subgroup of G M(G,A) Subgroup generated by <:I(g)g"1 where g e G, oeAEAut(G) Mi+1(G,A) M(Mi(G,A),A) where MO(G,A) = G G Chain of subgroups of the group G; G:G=G 2G 2°°°2G = 1 o l n Cn Cyclic group of order n Sn The symmetric group on n letters An The alternating group on n letters GL(n,p) General linear group of nonsingular nxn matrices over a field of order p SL(n,p) Special linear group of nxn unimodular matrices over a field of order p Stab(G) The group of stabilizing automorphisms of G S(G) The group generated by the stabilizing automorphisms of G PAut(G) The group of power automorphisms of G PStab(G) The group of power stabilizing automorphisms of G Pi A pi-Sylow subgroup of G G-[A] The semi-direct product of G with A BIBLIOGRAPHY 10. 11. BIBLIOGRAPHY A. Baartmans, Ph.D. Thesis, Michigan State University. R. Baer, "Die Zerlegung der Automorphismengruppe einer Endliche Gruppe durch eine Hallsche Kette." J. fur Reine und Angewandte Math. 220 (1964), 45-62. , "Supersoluble Immersion." Can. J. Math. 11 (1959), 353-369. C.D.H. Cooper, "On Power Automorphisms." Math. Z. 107 (1968), 335-355. P. Hall, "Some Sufficient Conditions for a Group to be Nilpotent." I11. J. Math. 2 (1958), 787-801. W. Hightower, Ph.D. Thesis, Michigan State University (1970). B. Huppert, Endliche Gruppe I, Springer-Verlag Inc. (1967). L. Kaloujnine, "Uber gewisse Beziehungen zwischen einer Gruppe und ~ihren Automorphismen." Berlin Math Tagung (1953), 164-172. P. Schmid, "Untergruppenreihen normalisierende Automorphismengruppen." Arch. Math. 23 (1972), 459-468. W.R. Scott, Group Theory, Prentice-Hall Inc., Englewood Cliffs, New Jersey (1964). H. Zassenhaus, The Theory of Groups, Chelsea Publishing Co., New York (1953)- 65 ‘ . 1 III H |[ i) (I T. lelllllEllN 010