' ' “ ‘ - 1‘. ‘F'x. 4 SOME SOLVABLE SUBGROUPS or V . . = THE OUTER AUTOMORPHISM. emu? ,. w~ » : f : . ‘ .OFAFINITE-GROUP ‘ ¥ _ - Thesis for the‘Degree of'Ph.V-D. ' ' 7 , MICHIGAN STATE UNIVERSITY * » , . _ . NGUYENCAODAMVi. .- , . «- r .‘h wwm, §s 35:13 fi 3 ~% W" ”Ir '5‘. ‘" ”1”“:9 ”ru- 1’30"“ W L l 1'; AA K Y Michigan State University , v This is to certify that the thesis entitled , SOME SOLVABLE SUBGROUPS OF THE OUTER 39¢ AUTOMORPHISM GROUP OF A FINITE GROUP presented by Nguyen Cao Dam n accepted towards fulfillment has bee of the requirements for Ph ,D . degree in Mathemat iCS Major profe or Date February 1, 1972 0—7639 l I" ABSTRACT SOME SOLVABLE SUBGROUPS OF THE OUTER AUTOMORPHISM GROUP OF A FINITE GROUP By Nguyen Cao Dam Our objective is to find some solvable subgroups of the outer automorphism group of a finite group G. In 1966, G. Glauberman [4] proved the following result: "Let G be a finite group having no non identity normal subgroups of odd order. Suppose $2 is a Sylow 2-subgroups of G. Let F[A(G);SZ] be the group of all automorphisms of G that fixes 52 as a set, and let F[A(G);82] be the group of all elements ofA(G) that fixes each element of S Then Out(G) 2. is solvable if and only if F[A(G);SZ]/F[A(G);Sz] is solvable." The idea is to focus our attention on the action of A(G) on a particular subgroup, namely 82. In Chapter I and Chapter II we try to find a similar result for an arbitrary finite group G. Let B* be a subgroup of Out(G), G is a finite group. Let B be the complete inverse image of 3* in A(G). In Chapter I we follow J.R. Weaver [14] and define a subgroup M to be B- intravariant if for each G in B there exists a g in G 01' such that M g = M, where Tg is the inner automorphism of G induced by g. Nguyen Cao Dam We define the following groups: F(B;M) = group of all a in B which leaves M fixed as a set F(B;M) = group of all g in B which leaves each element of M fixed R(B;M) = F(B;M)‘M = the restriction of F(B;M) on M NG(M) N04) = F(B;M) n I(G)\M a (TC—W E(B;M) = W . * _ We then show that B is solvable if and only if both R(B;M) F(B;M) F(B;M) fl I(G) are solvable. Thus if fifB;M) is solvable, and we do not necessarily have B* solvable. But for the class of finite solvable groups and the class of Frobenius groups, if we take M to be Fit(G) and kernel of G respectively, then EKB;M) solvable does imply 3* solvable. We then give an example of a Frobenius group with a solvable automorphism group. Results of Chapter I serve as basis for Chapter II. We can show that if foreveryprime p dividing \G‘, EfB;Sp) is solvable, then B* is solvable and conversely. The proof of this theorem uses the following known theorem of Dade that the group of all outer automorphisms which centralize every Sylow subgroup of G is solvable. In this thesis we give a proof of this theorem which is quite different from the original proof given by E.C. Dade [3]. From these theorems we get easily these following corollaries. Nguyen Cao Dam If A(Sp) is solvable for any prime p dividing ‘G‘, then Out(G) is solvable. PSL(2,p), p a prime, has a solvable outer automorphism group. We also generalize this following result by H. Liebeck [8]. .Any 2-generator nilpotent group G of order 2m3n has ea solvable automorphism group. We will no longer require G to be nilpotent. Instead we require that each S2 and 33 of G has at most 2 generators, the result will also hold in this case. In concluding Chapter II, we prove the following theorem: Let G be a finite group. Let 8* s Out(G), and let B be the complete inverse image of B* in A(G). Then the following three statements are equivalent. i) 3* is solvable ii) EKB;SP) is solvable forcmmnyprime p dividing ‘G‘ iii) EKBgNG(Sp)) is solvable foreweryprime p dividing ‘G‘ In Chapter III we follow another line in looking for solvable Subgroups of Out (G). Here we are given a chain 3: G = GO 2 G1 2...2 G 2 G = <1> of arbitrary subgroups of G. t-l t Let " e . 3(5): 30(3) = {e eA(G)\Gi = G1, 1 = O,l,2,...,t}. An automorphism a in 80(5) is said to stabilize the series 3 if [Ci-1’0] 5 G1 for i = 1,...,t. An automorphism a in 30(8) is said to locally stabilize the series 3 if for i = 1,...,t, there exists an inner auto- morphism Tg 6 80(3) 0 I(G) and depending on i such that i Nguyen Cao Dam [Gi_1,aTgi] 5 G1. A result of P. Hall [7] shows that any group of auto- morphisms of G which stabilize s is nilpotent. In this thesis, we show that, modulo the inner auto- morphism group, any group of automorphisms of G which locally stabilize the chain 3 is solvable of derived length at most t (t-I) + (2). As a side result, we follow A.D. Polimeni [10] to define recursively the elements of the stability series f g. Sk(s) = {e e sk_1(s)\[Gk_1,e] s G for 1 s k s t k} and also the elements of the local stability series. N = § Sk(s) {9 E k_1(s)l:arg 6 80(3) 0 I(G) such that [Gk-l’eTg] S Gk} for l S k S t. .Among the results we get, we cite the following ones. _ Sk(s) and §k(s) are normal subgroups of 80(3) for k = O,l,...,t. ._ = = 2 _... = If 31 G G0 G1 > 2 Gt—l 2 Gt <1> and 32 = G = H0 2 H1 2...2 Ht-l 2 Ht = such that G? = Hi Vi = O,l,...,t for a e E A(G) then -1 _ -1 ~ = ~ 9 [Sk(sl)]e — Sk(32) and e [Sk(sl)]e Sk(82)' ..If p is a prime dividing St(s) or §t(s), then p divides ‘G‘. _ If G is a finite solvable group, p the largest prime dividing \G‘ and s is a composition series for G, Nguyen Cao Dam then 80(8) is solvable and the p-Sylow subgroup of 80(8) is normal in 80(3) and is contained in St(s). - If G is a finite group and s is a chief series for G, then §t(s) = St(s) if and only if G is nilpotent. - If G is a finite group and s is a normal Hall chain g (s) then ‘jL-—- is nilpotent. I(G) SOME SOLVABLE SUBGROUPS OF THE OUTER AUTOMORPHISM GROUP OF A FINITE GROUP By Nguyen Cao Dam A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1972 FOR MY PARENTS ii ACKNOWLEDGEMENTS I am deeply indebted to Professor J.E. Adney for suggesting this investigation and for his helpful guidance during the pre- paration of the thesis. I am also grateful to him for all the things he has done for me during my entire Doctoral degree program, especially when struggling with the Agency for Inter- national Development in getting the extension of my financial support during my last academic year 1971-1972. My thanks are also due to the professors at Michigan State University who taught me Mathematics, especially those who introduced me to the beauty and challenge of Algebra. Last, but not least I wish to express my gratitude for my Vietnamese friends without whose encouragement and unfailing moral support this thesis could never be done. iii Chapter I II III TABLE OF CONTENTS INTRODUCTION 0.00.0.0.0...0.00.00.00.00...00...... ON B-INTRAVARIANT SUBGROUPS OF G ................ 1.1 B-Intravariant Subgroups ........... . ........ 1.2 Applications ......OOOOOOOOOOOCOO ........... O A NECESSARY AND SUFFICIENT CONDITION FOR A SUBGROUP OF OUT(G) SOLVABLE ... ..... .... ........ 2.1 Demonstration of the Theorem . ...... ......... 2.2 Applications .. .............................. 2.3 An Equivalent Formulation for Theorem 2.1.11 STABILITY SERIES AND LOCAL STABILITY SERIES ..... 3.1 Basic Definitions and General Properties ... 3.2 Some Properties of St(s) and St(s) ...... BIBLIOGMPIiY 0000000000 O 0 O OOOOOOOOOOOOOOOOOO O O O O 0 APPENDIX ........................................ iv Page 10 10 23 28 31 31 43 55 57 INDEX OF NOTATION 1. Relations: s is or be a subgroup of < is or be a proper subgroup of A. is or be a normal subgroup of is or be a proper normal subgroup of a: is isomorphic to E is an element of = is equal to C is isomorphic to a subgroup of II. Operations: X9 the image of an element x under the mapping 9 G6 the image of G under the mapping 9 Hx x-le f‘H restriction of the action of f on H % quotient group or factor group [x,y] x-ly-lxy the commutator of x and y [x1,x2,...,xn] [[x1,x2,...,xn_1], xn] X direct product of groups {x‘P} the set of all x such that P is true subgroup generated by all x such that P is true \G|, o(G) [G:H] |g\. 0(3) T 8 Groups: A(G) I(G) Out(G) Z(G) CG(H) NG(H) <1> F(B;M) F(B;M) R(B;M) number of elements in G (also called the order of G) index of H in G order of the element g the inner automorphism of G induced by g in G automorphism group of G inner automorphisms of G outer automorphism group of G center of G centralizer of H in G normalizer of H in G identity subgroup of a group a subgroup of Out(G) the complete inverse image of B* in A(G) the set of automorphisms in B which leave the subgroup M fixed as a set the set of automorphisms in B which leave each element of M fixed the restriction of the action of F(B;M) on M the restriction of the action of the inner automorphisms induced by NGCM) on M Rgeiug NCM) p-Sylow subgroup of a group G quotient group the symmetric group of degree n vi A n I G [G 90/] GL(n,q) SL (n 9Q) PSL Fit(G) O (G) CoreG(H) Hol(M;H) IV. Other 3 the alternating group of degree n the first derived group of G the subgroup of G generated by eflamec} group of n X n matrices of determinant * 0 over a field of q elements group of n x n matrices of determinant 1 over a field of q elements projective special linear group the Fitting subgroup of G the Frattini subgroup of G the maximum normal subgroup of G contained in H extension of M by H CZA(M) there is for any empty set set of all primes dividing \G‘ vii INTRODUCTION Our objective is to find some solvable subgroups of the outer automorphism group of a finite group G. In the first two chapters of the thesis, we let B* be a subgroup of Out(G), and let B be the complete inverse image of B* in A(G). We characterize the solvability of B* by looking at the action of B on a certain subgroup M of G or on the elements of the characteristic class of Sylow sub— groups of G. In Chapter 1, following J.R. Weaver [14] we call a sub- group M B-intravariant if for any element a in B there exists an x in G such that M“ = Mx. We show that if 3* is solvable then a corresponding factor group of a subgroup of A(M) namely R(B;M) must be solvable. The converse is not true. But for some classes of groups, for example, the classes of Frobenius groups and of finite solvable groups, if we choose M suitably the converse is then true. We then give an example of a Frobenius group with a solvable automorphism group. In Chapter 11, instead of fixing our attention to one particular group, we look at the action of B restricted on elements of the characteristic class of Sylow subgroups. Using a reSult by E.C. Dade [3] which says that the group of all outer automorphisms centralizing all Sylow subgroups of G is solvable, we can show that if R(B;Sp) is solvable for every prime p dividing ‘G‘, then B* is solvable. Together with the result in Chapter I, we have a criteria for deciding whether a subgroup ofOutflyis solvable or not. We can then use these results to prove the well—known fact that PSL(2,p) has a solvable outer automorphism group. We can also generalize a result by H. Liebeck [8] which says that any 2-generators nilpotent group of order 2m3n has solvable automorphism group. In Chapter III, we follow another line in looking for solvable subgroups of the outer automorphism group of a group G. Here we are given a chain 5: G = GO 2 G1 2...2 Gt—l 2 Gt = <1> of arbitrary subgroups of C. An automorphism a is said to stabilize the chain 5 if [Ci-I’d] s Gi for i = l,2,...,t. Any group of automorphisms of G which stabilize s is shown by P. Hall [7] to be nilpotent of class at most (:)‘ We now relax the condition [Ci-I’d] s Gi' An automorphism e is said to locally stabilize the chain 5 if for i = 1,...,t there exists an inner automorphism Tg. which fixes each Gi setwise and such that [Ci-1’9Tg,] s G:. We can show that any group of automorphisms which locally stabilize the chain 3 is solvable, modulo the inner automorphism group. Next following A.D. Polimeni [10], we define the stability series and the local stability series of the chain 5. The last term of the stability series and the local stability series are called St(s) and St(s) respectively. Given a finite group G and an arbitrary chain 5 of subgroups of G we can determine some properties of St(s) and §t(s) and also the relation between these two groups. The reader is asked to consult the index of notations for identification of symbolic notations of groups and relations. Further we have an appendix consisting of theorems which are referred to throughout the thesis. CHAPTER I ON B-INTRAVARIANT SUBGROUPS OF G In this chapter we will look at the action of a subgroup B of the automorphism group of a group G on some particular subgroups of G. In particular we will examine how the solvability of B ISG) I(G) influences the outer automorphisms of a certain sub- group of G and vice versa. The results in this chapter will serve as basis for Chapter II. Groups in this chapter will not be assumed to be finite unless expressly stated. Let B be a subgroup of the automorphism group of G, which contains the inner automorphism group I(G). Following J.R. Weaver [14] we define B-intravariant subgroups of G. 1.1 B-Intravariant Subgroups Definition 1.1.1. Let M be a subgroup of G. M is called B-intravariant if and only if for any 0 in B there exists OTg an Tg in I(G) such that M = M. We will also use the following notation: F(B;M) = set of g in B which leaves M fixed as a set. F(B;M) is clearly a subgroup of B. F(B;M) set of o in B which leaves each element of M fixed. F(B;M) is a subgroup of F(B;M). R(B;M) = F(B;M)‘M = the restriction of F(B;M) on N(M) = F(B;M) n I(G)‘M. N(M) is isomorphic to NGGM) . CGCM) F(B;M) F(B;M)[F(B;M) fl I(G)] Lemma 1.1.2. Let M S G. Then is RgBiM) isomorphic to N(M) 2:29;: We know that restriction is a homomorphism from F(B;M) onto R(B;M). Let n be the composite natural homomorphism from F(B;M) onto Bigifll- as given in the following scherne. N(M) F(B;M) restriction R(B;M) RéBéMZ \\\\\\\\\-n ,‘i,_-»///4 Then clearly Ker n = F(B;M)[F(B;M) fl I(G)]. Thus PM) 2. 8.91211 F(B;M)[F(B;M) fl I(G)] ”' N(M) Remark: Bé%fi%l is a quotient group of a subgroup of the auto- morphism group of M. Proposition 1.1.3. Let M be a B-intravariant subgroup of G. * Then B - — I(G) is solvable if and only if the following two conditions are satisfied: i) RngM! ' b N(M) 18 solva 1e F(B;M) F(B;M) fl I(G) is solvable. ii) Pro—of: Since M is B-intravariant, for any 0 in B there exists an 'Tg in I(G) such that 04g 6 F(B;M). Thus a E F(B;M)I(G). On the other hand, clearly we have F(B;M)I(G) S B. Hence B = F(B;M)I(G). Now 3* = B_ = F(B;M)IgG) 3 r0394) I(G) I(G) — F(B;M) fl I(G) FKBJM) F(B;M) fl I(G) * Thus B is solvable if and only if is solvable. Also since F(B;M)[F(B;M) fl I(G)] is normal in F(B;M) by F(B EDILQLML fl I(G); F(B;M) n I(G) Lemma 1.1.2, the quotient group is I“ (B ;M) F(B;M) fl I(G) normal in Taking the quotient of these groups I(B;M) F(B;M) n IIG) a, I(B;M) F(B;M)Ll‘iBm) n I(G)] ‘ F(B;M)[I‘(B;M) n I(G)] F(B;M) n I(G) we get The latter one is isomorphic to Bé%fi%l- by Lemma 1.1.2. Hence by the theory of solvable groups we see that 11B 3M) NE W n I(G) conditions are satisfied. . RSB;M2 . 1) N(M) is solvable 11) F(B;M)m‘asm) n I(G)1 F(B;M) n I(G) is solvable if and only if the following two is solvable. F(B;M) F(B;M) fl I(G) But the latter one is isomorphic to Remark 1: If TIC). has derived length less than or equal to n RngM) then so has . N(M) 2: For abbreviation we will use the notation R(B;M) R B'M ' t ad f # . ins e o N (M) 1.2 Applications By a complex M of G we mean any non empty subset of G. We have the following lemma. Lemma 1.2.1. Let M be a complex of G. If 0 is an element of A(G) which fixes M elementwise, then 0 induces the identity NG (M) cG (M) automorphism on Proof: NG(M) and CG(M) are well known to be subgroups of G. Clearly N (M) and C (M) are fixed by o as sets. So G G NG (M) o induces an automorphism on the quotient group C_IMI . We want to prove that n- no is in CGGM) for an arbitrary n in NG(M). Let m be an arbitrary element of M and n an arbitrary element of NG(M). By hypothesis we have 1)o o o -l (nmn-1)=(nmn- =nm(no) . Thus (nO)-1n m n-lnO = m i.e. Corollary 1.2.2. Let M be a characteristic subgroup of G with the property that the centralizer of M. in G is equal * to the center of M, then B = 13G) is solVable if and only if R(B;M) is solvable. Proof: M, being a characteristic subgroup of G, is trivially B-intravariant. By Proposition 1.1.2 we need only prove that F(BEM?;%)I(G) is always solvable. Since F(B;M) centralizes . N (M) NG(M) G M, F(B;M) centralizes ESTES. by Lemma 1.2.1. But CE? 3-: TIM; by hypothesis. So a fortiori F(B;M) centralizes fi'. F(B;M) centralizes both E: and M is well known to be abelian (see Theorem.N in the Appendix). Thus F(B;M) is always abelian. Q.E.D. F(B;M) fl I(G) Remark: Two classes of groups particularly suitable for applica- tion of Corollary 1.2.2 are: 1. Class of finite solvable groups G and we take M to be the Fitting subgroup of G (Theorem C in the Appendix). 2. Class of Frobenius groups G and we take M to be the kernel of G. i Example 1.2.3. 0 _ -1 Let x — [3 2] and y — [1 0] be two elements of GL(2,5). Let H = . Let M be elementary abelian of order 25. Let G = Hol(M;H) = the splitting extension of M by H with the inclusion isomorphism of H into A(M). Then A(G) is solvable. Peer: We will show that G is a solvable Frobenius group with kernel M. By using Corollary 1.2.2 we will show next that Out(G) is solvable. Hence A(G) is solvable. Now by an easy computation we see that o(y) = 6, x-lyx = y-l, and x2 = y3 Thus H is the extension of a normal subgroup of order 6 by a group of order 2. Also by computing the characteristic polynomial of each non-identity element of H we see that no non-identity element of H has characteristic value 1. Thus for any1fx E M the centralizer of x in H is the identity group. So G is a Frobenius group with kernel M. G is solvable since it is an extension of the abelian group M by the solvable group H. On the other hand A(M) = GL(2,5) has order 25 X 3 X 5 while N(M) 35H has order 22 X 3. Thus R[A(G);M] = RI A(G) mi N (M) theorem [6] (Theorem B in the Appendix) R[A(G);M] is solvable. must have order dividing 23 X 5. By Burnside By Corollary 1.2.2 Out(G) is then solvable. Q.E.D. CHAPTER II A NECESSARY AND SUFFICIENT CONDITION FOR A SUBGROUP OF OUT(G) SOLNABLE In Chapter I we have shown that if B* is a solvable sub- group of the outer automorphism group of a finite group G, and if we consider the complete inverse image B of 8* in A(G) , then RIB38p) is solvable for any p-Sylow subgroup Sp of G. A natural conjecture arises. "If for every Prune P dividing the order of G, R(B,Sp) is solvable, then B* is solvable." We will show in this chapter that this is true and derive some consequences from the theorem. In particular we can use the theorem to generalize the following result of H. Liebeck [8]. "If G is a nilpotent 2-generator group of order 2m3n then Aut G is solvable." All groups in this chapter will be assumed to be finite. 2.1 Demonstration of the Theorem The proof of the theorem depends on this result by E.C. Dade [3]: "The group of outer automorphisms centralizing all Sylow subgroups of a finite group G is solvable." For the sake of completeness, a proof of this theorem - which is different from the original one, and we believe it is simpler - is given here. 10 11 We give some definitions which are from E.C. Dade [2]. An outer-automorphism w of a finite group G is a coset of the normal subgroup I(G) of all inner automorphisms in the group A(G) of all automorphisms of G. Definitions 2.1.1. We say that w centralizes a subset S of G if there is some automorphism o E m such that x0 = x, for each x E S. We call w locally trivial if it centralizes, in this fashion, every nilpotent subgroup of G. The set LT(G) of all locally trivial outer automorphisms of the group Out(G) of all outer automorphisms of G has some nice properties, as the following theorem shows. Theorem 2.1.2. (First Dade theorem [2]). If G is a finite group, then LT(G) is a normal nilpotent subgroup of Out(G). Now if instead of requiring w to centralize every nilpotent subgroup of G we just require w to centralize every Sylow subgroup of G then we get the following theorem. Theorem 2.1.3. (Second Dade theorem [3]). For a finite group G, the set SC(G) of all outer automorphisms centralizing all Sylow subgroups of G is a normal solvable subgroup of Out(G). The proof of this theorem depends on this following lemma, also by E.C. Dade [3]. Lemma 2.1.4. Let B1"°°’Bn (n 2 1) be groups and C be a subgroup of the direct product B1 X...X Bn. Let D be a normal subgroup of C such that pri(D) = Bi’ for 12 i = 1,...,n where pri denotes the natural projections of B X...X B on Bi° Then %' is nilpotent of nilpotency class If n = 1, then C = D = B, and the lemma is true (of course we consider the identity group as nilpotent of class 0). So we may suppose that n > 1, and that the lemma is true for any value less than n. Case 1: Suppose B1 0 D = <1>. Let pr2 be the natural projection of ’00., ... .0. O X... B1 X X Bn onto B2 X X Bn Then clearly B2 X Bn’ (C), pr2 n(D) satisfy the hypothesis of the ’00.,“ ’00., Prz lemma with n-l instead of n. By inductive hypothesis pr2 (C) LOCO; Pr2 n(D) ’00., is nilpotent of class < n-l. Now let n be the natural homomorphism from C onto pr2 n(C) "°°‘* as given in the following scheme pr2 (D) , O O O ,n C pr2,...,anr (C) canonical_ pr2,...,n(C) \L”. ,n/ pr23‘°°9n(D) T] , Then clearly Ker n = = (B1 fl C)D. Thus TEE—[7575' is nilpotent of class < n-l as being pr mm)- isomorphic to (D) . 0.0, C 2, P132 , 9. But ._._._D—....._._ . o h. (B1 0 C)D lS isomorp 1c to yglr?r636' D 13 So to prove that g' is nilpotent of class < n it (B r10)D C suffices to prove that --jE--'; 2(5). But the groups B1 0 C and D are both normal in C and theirintersection B1 H C 0 D = B1 0 D = <1> by hypothesis. So B1 H C centralizes D. Since projections are homomorphisms, we have B1 0 C = pr1(B1rlC) centralizes pr1(D) = B1. Thus B1 0 C is contained in the center of B X...X Bn. It follows 1 (B n c)D D l C that is contained in the center of DI' Case 2: B1 0 D # <1>. Let b 6 B1 H D and b E B 0 Since pr1(C) 2 pr1(D) = B1 1. 3x 6 C such that pr1(x) = b. ‘Now D is normal in C so we x b x . have b0 6 B1 0 D. But bO — b0 6 Bl. Thus B1 H D 18 normal in B1. Hence B1 0 D is normal in B1 X...X Bn. We can form B1XOOOXBn B X...X Bn Bl H D . But B1 0 D C B1, so we can identify B1 n D Bl with -————-— X B X...X B . Also in the process of identifica- B1 H D 2 n tion we identif -"—9——' and ———2—- with C' and D' two ’ y B On B no ’ l 1 B1 x 0 subgroups of Bl m D B2 X...X Bn’ respectively. Clearly B l BEIFFB.X B2 X"°x Bn’ C', D' satisfy the hypothesis of the lemma B with the additional property that 11' fl g—-%j6 = <1>. By Case 1 l c' C . . DTIE 5- is nilpotent of class < n. Q.E.D. 14 Proof of Theorem 2.1.3: Suppose that SC(G) is not solvable. There exists a subgroup 5* with the property that its derived group 1 + (5*)' = 3*. Let S be the complete inverse image of 8* in A(G). A subgroup R of A(G) is said to £2235_ S if S = R I(G). Also if M is a subgroup of G and S a subgroup of G then F(S;M) denotes the subgroup of S which centralizes any element of M. * We will show that S is contained in LT(G), i.e. we show this following assertion: ”Any element of 8* centralizes every nilpotent sub- group of G." Let N be a nilpotent subgroup of G. N is the direct product of its Sylow subgroups Pi N=P. X...XP . (1) If n = l, the assertion is true by definition of SC(G). So suppose that n > 1, and suppose that the assertion is true for any integer less than n. We may also suppose that N is maximal in the class of all subgroups of G having the form (1) i.e. same number n and same set of primes dividing ‘N‘ Let Pi = P x...x Pi x P x...x P for i = 1,...,n. Then 1 -l i+l n Pi is a Sylow subgroup of CG(Pi), for otherwise N would not be maximal. Let n be the canonical homomorphism of G onto i I(G). Then “(Pi) is a Sylow subgroup of “(CG(§1)) = F[I(G);Pi] = F(S;Pi) n I(G). 1 15 Now F(S;Pi) covers 8 by inductive hypothesis. Let a e F(S;Pi). We have [n(Pi)]° : F(S;Pi) n I(G) = F[I(G);Pi]. Since F[I(G);Pi] is normal in F(S;Pi), applying the Frattini argument we know there exists an T in F[I(G);Pi] such that [N(Pi)]o = [n(Pi)]T§. Thus -1 ., (fig 6 NS(T\(P1)) fl F(S,Pi) - (2) We claim Ns(fi(Pi)) = F(S;Pi) where F(S;Pi) denotes the set of elements of S which fixes Pi as a set. First, we have r(s;Pi) g Ns(n(Pi)) because T: = Txa for a E A(G), and Tx E I(G). On the other hand if y E NS(n(Pi)), then [PiZ(G)]Y = PiZ(G), which clearly implies that P: = Pi because PiZ(G) is a nilpotent group and Pi is a Sylow subgroup of PiZ(G). Thus yer(s;Pi). So F(S;Pi) =NS('n(Pi)) as claimed. Now we replace NS(n(Pi)) by F(S;Pi) in (2) and we get 0 e [F(S;Pi) O F(8 $51)] PING) £51]. So, so far we have shown that: F(S;Pi) s [F(S;Pi) n F(S;Pi)] F[I(G);Pi]. The other way around is trivial, so we get F(SsPi) = [F(S;Pi) fl F(S;Pi)] Film) £1] . This equation shows that: Ir(s;Pi) n F(S;Pi>]I(G> = s (3) l6 i.e. F(SgPi) n F(S;Pi) covers S . (3') Now we have A(N) = A(Pl) X...X A(Pn)’ because each Pi is characteristic in N. Let R(S;N) be the restriction of the action of F(S;N) on N and let R(I(G);N) be the restriction of F(I(G);N) on N. Then since F(I(G);N) is normal in F(S;N) we have R(I(G);N) A_R(S;N). Also let Ai be the projection of R(S;N) on A(Pi)° Then we see that R(S;N) is a subgroup of A1 X...X An. Next, because of (3) we have: F(SsN)= [F(S, P i) D F(Sé i)]F(I(G) ;N) V i = 1,...,n So v i = 1,.--,n. pri(R(S;N)) = Pri(R(I(G);N)) = By Lemma 2.1.4 where we let C = R(S;N) and D = R[I(G);N] we get R%I%G§%N] is nilpotent. (4) But R(S;N) I(S gN) RII(G);N] F(s ;N)r[I(G) ;N] 2. F(SgN) //’Fj§;N)F[IiG)iN1 . "F(I(G);N) F(I(G);N) F(83N2 F(S ;N) 3F(S ;N)I(G)_ . And because F(I(G);N) =F(S N) O I(G)_I(G)-S*wh1ch has the property that (S *)' = 8*, the quotient group fé%?%%—fi3y/’F(S;§;Eg§$§%iN1 must have the same property as that R(S;N) of 8* . Therefore must also have that same ro ert . R[I(G);N] p p y 17 Together with (4) we conclude that R(S;N) R[I(G);N] = <1> . s0 mm) = F(S;N)r(1(c);m F(I(G);N) F(I(G);N) which implies that F(S;N) = F(S;N)P(I(G);N). But F(S;N) 2 F(S;Pi) n F(S;Pi) and the latter one covers S. So F(S;N) covers S, and F(S;N) also covers 8. This proves the assertion. We have now proven that 8* is a subgroup of LT(G). Since LT(G) is nilpotent by Theorem 2.1.2. This is a contradic- tion which proves that SC(G) is solvable. Normality of SC(G) in Out(G) is trivial to verify. Q.E.D. Now before proving the theorem announced at the beginning of the chapter we need some more preliminaries. Let G be a finite group and Sp be a p-Sylow subgroup of G. Let Ap be a subgroup of the automorphism group of Sp Consider the set E(Ap) = {a e A(G)] 3 TX 6 I(G) such that O’Txlsp 6 AP} Lemma 2.1.5. E(Ap) is a subgroup of A(G), and I(G) A_E(Ap). Proof: Let a and B be two arbitrary elements in EQAP). There exist T and TB belonging to I(G) such that E A St G‘s P B‘s P P P 18 Since restrictions are homomorphisms we also have -1 -l = A “TO(BTB) \S (aTa‘S )((BTB) ‘S ) E p P P P B t ( )-1 = ‘1 -1 = '( '1 8.1 U aTa 8TB aTaTB 6 a6 TQTB ) = QB‘HZ (for Tz E I(G), since I(G) AA(G)). So UB-ITZ‘ E AP. This implies that asdlis an element of S P E(Ap). The fact that I(G) is contained in E(Ap) is clear. Q.E.D. Lemma 2.1.6. If Ap is normal in the automorphism group A(Sp) of Sp’ then E(Ap) is normal in A(G). Proof: Let a E E(Ap) and 9 E A(G). By Sylow theorem, there -1 exists a z E G such that SE S: . This implies that Or EAS . zls ( P) P Since a E E(A ), 3 T E I(G) such that am 6 A . p a 0‘s P P By hypothesis Ap is normal in A(Sp), so we have GTZ‘ ) 3" - A . (Cl/Ta‘s C P P Again restrictions are homomorphisms, so we have 9T Z (as ) e A 0’ 19 By a similar computation as in Lemma 2.1.5, we get 672 9 (ma) = (oz )Tu fora Tu in I(G) 9 This means that a E E(Ap). Q.E.D. Definition 2.1.7. Let n be the set of primes such that each p E n divides the order of G. let Ml be the collection of subgroups Ap of A(Sp) -—- one for each p E n - such that each Ap contains N(Sp) and is contained in the normalizer of N(Sp) in A(Sp)’ i.e. = A :m {Ap‘ p sA(Sp) and N(Sp) QAp}p e n (Recall that N(Sp) is the restriction of the action of N(S) I‘(I(G);S ) on S . N(S ) is isomorphic to 5—P— ). P P P C (S ) G p A We call an solvable if the quotient group N-(g—I is P solvable for every p in 1'1. If Sm is solvable, we define the derived length rim of gm to be the maximum in the set of all derived lengths of A may ’ P E “- P By convention the derived length of <1> is 0. Now let E(SJI) = fl E(Ap) oSince I(G) QE(AP) for PErT V p E 11', we also have I(G) AECDI). Thus we can form Km) =E—@3l. I(G) Proposition 2.1.8. If every Ap in 931 is normal in A(Sp) 7': for p E n, then E EDI) is normal in Out(G). 20 Proof: Lemma 2.1.6 shows that E(Ap) is normal in A(G) for any p in 11. Thus E011) = fl E(Ap) is normal in A(G). En .. . E4211. p. A491 . OEOD. This implies I(G) is normal in I(G) Q Proposition 2.1.9. If SC(G) has derived length n1 and El * is solvable of derived length n2, then E (m0 is solvable of derived length at most n1 + n2. If n2 = 0 then Ap = N(Sp) for any p E n. It is easy to verify that the definition of E*€m) in this case coincides with the definition of SC(G). So 'E*Cm) is solvable of derived length n1 = n1 + 0. So suppose n2 > O, and suppose the theorem is true for all sets T1 with ncm).< n2. Let a and B be two arbitrary elements of E(mb. There exist Ta(p)’ T (p) in I(G) depending on p E n such 8 that GT (P) E A and T ( ) E A P P I Now ,T() EA,A =A. [aTa(P)‘S B B P ‘S 1 I p p] p P P By a similar computation as in Lemma 2.1.6, this implies that * u ( ) [Q’BJTz‘sp 6 AP for T2 6 I(G) Now let Ufi'== A'N S . Then clearl { p ( p)}pEn y 21 N S A'N S S.A S . ( p) A P ( p) ( p) Thus EDI' has the same property as SDI. It is easy to check that fiR' is solvable of derived length at most n-l. By (*) we see that [01,8] E E(fifi'). So _ _ * (**) [01,8] 6 E 051') .. ~k _ where a = aI(G) E E (313!) and B = BI(G) are two elements of * E (50!)- 7% Relation (**) shows that the derived group [E (9]!)]' a: -k of E (EDI) is contained in E (EDP). By inductive hypothesis, the latter one is solvable of derived length at most n1 + n2-l. * So [E Cfil)]' is solvable of derived length at most 1: n1 + n2 - 1. Therefore E 0)!) is also solvable of derived length at most n2 + n1. Q.E.D. Corollag 2.1.10. If the set ‘11! defined above is solvable with the additional property that VIAp E Em, Ap is a normal sub— group of A(Sp), then E*(£D'D is a normal solvable subgroup of Out (G). Proof: Propositions 2.1.9 and 2.1.8. Using Proposition 2.1.9 we can give a criteria for solvability of a subgroup of the outer automorphism group of G. * Theorem 2.1.11. Let B be a subgroup of the outer automorphism * group of G. Let B be the complete inverse image of B in * the automorphism group A(G) of G. Then B is solvable if and only if R(B;Sp) is solvable for everyp in n. 22 2:22;: The "Necessity" part is Proposition 1.1.3. For the "Sufficiency" part, suppose given R(B;Sp) = R(B;S ) T9_f_= The "Necessity" part does not require SC(G) to be trivial. To prove it we just follow the proof of Proposition 1.1.3. The "Sufficiency" part does require SC(G) trivial. The proof of it is an adjustment of the proof of Proposition 2.1.9. 23 2.2 Applications We give some applications of Proposition 2.1.9 and Theorem 2.1.11. Corollary 2.2.1. Let G be a finite group. Out(G) is solvable if and only if R(A(G);Sp) is solvable for evetyp E n. P_r90_f.= Clear from Theorem 2.1.11. Definition 2.2.2. Let G be an automorphism of a group G. o is called a power automorphism if for any g in G, 0 maps g to one of its power. It is equivalent to the following: o is a power auto- morphism if and only if G fixes the cyclic group generated by g as a set, g being an arbitrary element in G. Corollary 2.2.3. Let G be a finite group. Let B be the set of all elements a in A(G) with the property that for any p-Sylow subgroup Sp, there exist an T (p) depending on a p E n such that CT (p) restricted to Sp is a power auto- a = B I(G) morphism of Sp. Then B* is a normal solvable sub- group of Out(G). m= Let Ap be the set of all power automorphisms of the p-Sylow subgroup Sp of G. Ap is a normal subgroup of A(Sp) because 1) Let a E.Ap. Since a is onto, ‘Vy E Sp, 3x E Sp such that xa = xn = y. a is 1-1 so o(x) and n are 24 relatively prime, i.e. = . This implies (xn)m = x Thus -1 a m Y = X = y -1 SO a E A . P 2) Let a E A and B E.A . Let x E S . P P P XaB = (Xn)m = nm So aBEA. 3) Let a E A and e E (Sp ). Let x E S . P _ P 9-1 a 6-1 n n 9 1 (x ) = (x ) = (x ) since a is a power automorphism. Thus x6 a = x . So Ap is normal in A(Sp). Now we claim that Ap is abelian. For let 0,3 be two elements of AP. Let x be an arbitrary element of Sp, since both a and B induce automorphisms on the finite cyclic group generated by , and because the automorphism group of a finite cyclic group is abelian we have XEQ,B] = x for x E S , i.e. P [a,B] = identity on Sp Thus if = {A MN(S?)3€" then ‘3)! satisfies the hypothesis of B* Corollary 2.1.10. 80 E* (SR) is solvable. Clearly E* (SIR)= Q.E.D. Corollary 2.2.4. If the automorphism group A(Sp) of a p-Sylow subgroup SI) of G is solvable for QVGryTp in n, then Out(G) is solvable. 25 Proof: v p e n, R(A (G) ;sp) is solvable. So R(A(G);Sp) is solvable as quotient group of a solvable group. By Theorem 2.1.11 Out(G) is solvable. Corollary 2.2.5. Let G be a finite group with cyclic Sylow subgroups. Then A(G) is solvable. trees: G is known to be solvable (see D.S. Passman [8]). The automorphism group of a cyclic group is abelian. By Corollary 2.2.4 Out(G) is solvable. Thus A(G) is solvable. Q.E.D. Now let p be an odd prime and let q = pn for n a positive integer. Consider GL(2,pn) the general linear group of 2 X 2 matrices over the Galois field of pH elements. We want to apply Corollary 2.2.4 to show that the projective special linear group PSL(2,p) has solvable outer automorphism group. Certainly this result is known by other method. First we quote some known results. Theorem F (see D. Gorenstein [5]). The following hold: i) SL(2,q) contains cyclic subgroups of order q-l and q+l. ii) A 2-Sylow subgroup 82 of SL(2,q) is generalized quaternion. Theorem G (see D.S. Passman [8]). Let G be a generalized quaternion group then the follow- ing is true: 26 i) if \G‘ = 8 then A(G) is isomorphic to the symmetric group of degree b ii) if ‘G‘ 2 16 then A(G) is a 2-group. From these two theorems and Corollary 2.2.4, we have the following proposition. Proposition 2.2.6. If p is an odd prime then SL(2,p) has (a solvable outer automorphism group. Press: SL(2,p) has order (p2-1)p. Now p and (pz-l) are relatively prime, so the p-Sylow subgroup Sp is cyclic of order p. A(Sp) is therefore cyclic. If r is a prime different from p and dividing (p2-1)p, then r divides p2-1 = (p-l)(p+l). If r divides both (p-l) and (p+l), then r divides 2p. Since r # p, r must be equal to 2. The 2-Sylow subgroup 82 is then a generalized quaternion group by Theorem F. By Theorem G, A(Sz) is solvable. If r # 2, then r divides either (p-l) or (p+l) but not both of them. Again by Theorem F, the r-Sylow subgroup Sr is cyclic. A(Sr) is then cyclic. Thus for any r\(p2-l)p, A(Sr) is solvable. By Corollary 2.2.4 Out(G) is solvable. Q.E.D. Proposition 2.2.7. If p is an odd prime then the projective special linear group PSL(2,p) has solvable outer automorphism group. Pear: Similar to the proof of Proposition 2.2.6. 27 Proposition 2.2.8. Let G be a finite group of order n n ‘G‘ = p11,...,pnn with ni s 3 and satisfying the following conditions: i) if pi is odd then the pi-Sylow subgroup Sp is i 2 either cyclic or an extra-special pi-group of exponent pi. ii) if the 2-Sylow subgroup S has 8 elements, then 2 it is non abelian. Then Out(G) is solvable. For pi = 2 the 2—Sylow subgroup 82 can be cyclic or Klein-4-group, or Dihedral of order 8 or quaterion of order 8. Each of these groups has a solvable automorphism group (see Theorem H in the Appendix). For pi # 2, A(Sp ) is cyclic if S is cyclic. For . p. i i 2 the case where S is extra-special of exponent p,, A(S ) . i p. i i is solvable by Theorem J.in the Appendix. Q.E.D. Proposition 2.2.9. (Generalization of Liebeck Theorem [8]). Let G be a finite group of order 2m3n such that each 2- Sylow subgroup S has at most 2 generators, and each 3-Sylow 2 subgroup 83 has at most 2 generators. Then A(G) is solvable. Proof: By Burnside theorem G is solvable. We need to verify that A(S and A(S3) are always solvable. Then we apply 2) Corollary 2.2.4 to conclude that Out(G) is solvable. If 82 (or S3) has one generator, i.e. if 82 (or 83) is cyclic, then A(SZ) (or A(S3)) is abelian. So let S be a two generator 2-group or 3-group. In Theorem I in the Appendix put N = Q the Frattini subgroup of S. Since §_ is elementary abelian of order 4 or 9, A(%) is Q 28 solvable. Hence éégl- is solvable, where Ml==A1(G;N) is the group of all automorphisms of G which fix g' elementwise. Since Em is a nilpotent group (Theorem P in the Appendix), A(S) is solvable. Q.E.D. 2.3 An Equivalent Formulation for Theorem 2.1.11 We have given a criteria for a subgroup 3* of Out(G) to be solvable by looking at its action on each p-Sylow sub- group SP of G. If B is the complete inverse image of B* in A(G), we have denoted by R(B;Sp) the quotient group R(B;S ) -—-JL- Notice that both R(B;S ) and N(S ) are subgroups N(S ) P P P of A S . (p) * Now if we look at the action of B on the normalizer NG(Sp) of Sp in G, then we show that we will have a similar result. First notice that for any 9 in A(G) we have by Sylow theorem 9 g s = s ) a G ( p) ( p g 6 which clearly implies that 9 = g Thus NG(Sp) is A(G)-intravariant. Also if g is an element of A(G) which fixes Sp, then 0 fixes NG(Sp) and conversely. So we have 29 F(stp) = I‘IBsNG(Sp)]. Also by Sylow theorem we have With these remarks we have the following theorem. * Theorem 2.3.1. Let G be a finite group and let B be a subgroup of Out(G). Let B be the complete inverse image of * B in A(G). Then the following three statements are equi- valent. * i) B is solvable ii) R(B;Sp) is solvable for everyp in n iii) R(B;NG(SP)) is solvable for everyp in Tr. Em}? i) and ii) are equivalent by Theorem 2.1.11. i) = iii) by Proposition 1.1.3 where we replace M by N (S ). G p _ R(BsNG(sP)> iii) 2 ii) we have R(B;NG(SP)) = N(NG(SP)) which by Lemma 1.1.2 is isomorphic to F B;N S B;S ( G

> r< B> F(B;NG(SP))[F[B;NG(SP)] fl I(G)] — F(B;NG(SP))[I‘[B;SPJ n I(G)] since F(B;Sp) = F(BgNG(Sp)). B t clearl F B;N (S S F B;S , so u y(Gp))(p) 3O F(B;NG(SP))[F(B;SP) n I(G)] s F(B;Sp)[F(B;Sp) n I(G)] SO F(B;NG(SP)) R(B'S ) = F(B;sp) _-EEF(B;NG(SP)[P(B;NG(SP)) n I(G)] ’ p F(B;Sp)[F(B;Sp) n I(G)] F(B;Sp)[F(B;SP) n I(G)] F(B;NG[r n I(G)] So R(B;Sp) is solvable being a quotient group of a group isomorphic to the solvable group R(B;NG(SP)). Q.E.D. CHAPTER III STABILITY SERIES AND LOCAL STABILITY SERIES In Chapter II we have given a criteria for a subgroup B* of Out(G) to be solvable in terms of its action on the p-Sylow subgroup Sp of G for any prime p dividing the order of G. In this chapter we will follow another line in trying to find some solvable subgroups of Out(G). This work is influenced by the work of P. Hall [7] on the automorphisms stabilizing a chain of subgroups of a group G. We will also follow some ideas of A.D. Polimeni given in his thesis '% Study of the Automorphisms and Chains of a Finite Group" [10]. Groups in this chapter will not be assumed to be finite, unless expressly stated. 3.1 Basic Definitions and General Properties Definition 3.1.1. Let s: G = G 2 G 2...2 G 2 G = <1> be a chain of subgroups for a group G with G1 being arbitrary in Gi-l as well as in G for i = 1,...,t. _ 9 _ . _ Let 30(5) — {e e A(G)\Gi — G1 1 — O,l,...,t}. 80(8) is clearly a subgroup of A(G). An automorphism a in 80(8) is said to stabilize the series 3 if [Ci-1’0] is a subgroup of G1 for 31 32 i = 1,...,t. ([Gi_1,a] denotes the subgroup of G generated by the -1 a set is s Is 6 Gi_1].) An automorphism a in SO(s) is said to locally stabilize the series 3 if for i = 1,...,t there exists an inner automorphism Tg- belonging to S0(s) fl I(G) and L depending on i such that [G. ,aT _] is a subgroup of Gi‘ 1']. £1 Let H be a subgroup of automorphisms of G which locally stabilize the series 5. Let H be the set of all automorphisms of G which stabilize the series 5 and is contained in H. Then we have the following result. Lemma 3.1.2. H is a nilpotent normal subgroup of H. Proof: a) H is a subgroup of H. Let a E H and g E Gi’ -l -1 then we have g‘lga = [(g-lga)-l]a which belongs to Gi by definition of H and 80(5). So a-1 E H. Now let a and B be two elements of H. Then for g in Gi we have -1 -l -1 g go’E5 = g ga(g“) (ga)B 6 Ci- SO as EH. b) H is normal in H. Let 9 be an element of H e and let a be in H. If g E Gi then we have g E Gi-l -1’ by definition of 80(5). This implies that = g x for some x E Gi (by definition of H). 33 -1 -1 6 a9 -1 Hence 8 = (g e e -1 e as = x9 x) = gx . So g g E Gi' c) H has been shown to be nilpotent of class at most (5) by P. Hall [7]. Now because I(G) is normal in A(G), 30(3) 0 I(G) is also normal in 80(3). We can form H[So(s) fl I(G)] and H[So(s) fl I(G)]. Since H is normal in H we can also form ~* fi[so(s) n I(G)] ' Htso n I(G)] ~* Concerning H we have the following result. Proposition 3.1.3. Let G be an arbitrary group and s: G = G0 2 G1 2...2 Gt-l 2 Gt = <1> be an arbitrary chain of ~* subgroups of G. Then the group H defined above is nilpotent of class at most t - 1. Proof: Let 91,...,er be r elements of H. There exist T ,...,T in 30(8) 0 I(G) such that [G0,61Tg1] 3 G1 t t t Then we claim that the commutator [e T ,...,e T ] is an 1 s1 t at element of H, i.e. we want to prove that for g E Gi-l’ [81Tg1,...,etrgt] g = gx (for some x E Gi) and this for any 34 For abbreviation we let 9; = eiT . Define Y1 = 9 and recursively = ' '= - vi“ [yi,ei+1], 1 l,...,t 1. -1 Let be an element in G Then Yi-l ‘ G g y 1‘1. g 6 1']. because Yif S0(s). Hence -1 e: -1 (gYi-i) l = gYi- x (for some x E Ci) by property of 9;. So v-1 9'v v’1 Y g 1-1 L i-l = (g i-lx) i-l = g XYL£ Therefore I 1 I ' [Yr-1’91] _ Y1-1 91 _ 91 Yi-iei g - (gx ) -g X I 'Y._ . = gyx 1 1 1 with y 6 Ci. i.e. H. .63] Y- 93 g 1-1 1 = gz with z = yx 1-1 1 E Gi’ i.e. for any g in Gi-l’ there exists a z in Gi such y. that g 1 = gz, which in turn implies that —l -1 -1 (Y- ) _ v. _ Y- gl =g(zl)1 with (zl)lEG. Thus 35 e! ' '1 "1 i+1 Yi+l [Yi’ei+l] _ . Yi 91+1Yi\ 8 = 8 ' 8 '1 "1 ' y. e. e. = (g 1 +1 u) 1+1 for some u E G1 -1 ' __ Y1 91+1 -1 -1 i 9i+1 = g(z ) u = gv with f1 ' V = 2.1) 1 1+1 E G. 1 So proceeding similarly for the next index we see that Yt g = gw for some w E Gi' But since Tg E 80(5) fl I(G) A_S0(s) we have i vt = [911g "°°’9tTg 1 = [91....,et]Th l t for some Th E S0 fl I(G) that means mod(H[SO(s) fl I(G)]) [91:--°39t15 1 ° ~* This shows H is nilpotent of class at most t-l. Q.E.D. Corollary 3.1.4. Let s: G = G 2 G 2...2 G Gt = <1> O 1 t-l 2 be an arbitrary chain of subgroups of G. Let H be a group of automorphisms of G, which locally stabilize the series 3. Let H be the largest subgroup of H which stabilizes the series 3. Then the following two statements hold. 36 i) ELISE). is nilpotent of class at most t-l H I(G) ii) :—Tfl-- is solvable of derived length at most H n I(G) E (t‘l) + (2) Proof: 1) An element of H I(G) has the form eTg where e is in H and Tg is in I(G). Let 91Tx_,...,eth be t elements of H I(G). Since 1 t I(G) is normal in A(G) we have (1) [61Tx1,...,etTXt] = [91,...,et]TZ for some Tz E I(G). By the proof of Proposition 3.1.3 we know that [91,...,et] a i mod[H[SO(s) n I(G)]] 2 '. . ... = ° ( ) 1 e [91: ,et] QTU With a E H and Tu E SO(S) fl I(G) (l) and (2) together shows that Q’T T ... T = [91 x ’ ’et xt] OZTuTz uz l with a Tuz E H[SO(S) fl I(G)]I(G) H I(G). So [9 T ,...,e T ] E 1 mod H I(G) which means that 1 x1 xt ~ H.119). is nilpotent of class at most t-l. H I(G) H I(Gzzk H We have I(G) __ liq—fl I(G) But H is nilpotent of class at most (5) by Theorem ii) N in the Appendix. 37 Now H I(G) __fi I(G) HI(G) H I(G) I(G) I(G) So EL%{%%- is solvable of derived length at most (t-l) + (5) since it is extension of a nilpotent group of class at most (5) by a solvable group of derived length at most (t-l). ~ And so is ——H—— Q.E.D. H n I(G) A natural question arises "Is E-%%g%- nilpotent?" We could not answer this question in the general case. Corollary 3.1.5. If 3: GO 2 G1 2...2 Gt-l 2 Gt = <1> is a normal series of G (i.e. G1 is normal in G for i = 1,...,t) then I(G) is contained in 80(3). ~ N If H contains I(G) then IEG) is solvable. Use Corollary 3.1.4. In his thesis A.D. Polimeni [10] has defined the stability series of a subnormal chain 3 of subgroups of G and has studied some of its prOperties. In a similar manner we now define the stability series of an arbitrary chain 3 of subgroups of G. We also define the local stability series of s. We will then study some prOperties of these two series. ' ' ' . . . : = 2 2...; 2 = 1 Definition 3 l 6 Let s G G0 G1 Gt-l Gt < > be a chain of arbitrary subgroups of G. We define _ ~ _ 9 _ . _ 30(s) - 30(s) - {e e A(G)\Gi — Gi for 1 — O,l,...,t] and recursively 38 Sk(s) = {e e sk_1(s)l[Gk_1.ej s Gk} for l S k s t and also recursively Sk(s) = {e E §k(s)‘:aT§ E 80(3) fl I(G) such that [Gk_1,9Tg] 5 Gk} for l s k S t. (Notice that in the definition of §k(s), the element Tg depends on e and k also.) Clearly S0(s) = 80(5) is a subgroup of A(G). For the other set Sk(s) and Sk(s) we have the following properties. Proposition 3.1.7. Let G be an arbitrary group and s: G = C0 2 G1 2...2 Gt-l 2 Gt = <1> be a chain of arbitrary subgroups of G. Then i) Sk(s) is a normal subgroup of 80(3) ii) Sk(s) is a normal subgroup of §0(S) We prove that Sk(s) is a normal subgroup of 80(5) for k = O,l,2,...,t. For k = 0 it is trivially true. So suppose 81(5) is a normal subgroup of 80(3) for i < k. a) Sk(s) is a subgroup of SO(s). if a is an element of Sk(s), there exists an element Ta in 30(3) 0 I(G) CIT such that if g is in Gk-l we have g-1g a E G k’ i.e. 39 GT g a = gu for some u E Gk -l -l -1 (OT ) (OT ) (OT ) so g=(gU) a =g a u 0’ -l -l -l (aTa) _1 (OT ) _1 (aTa) which implies that g = g(u ) a with (u ) E Gk' But (aT )01 = T-la-l = 0—1T' for some T' E S (s) H I(G), a a a a 0 i.e. a-1 E Gk. Now let a and B be two elements of §k(s). There exist Ta and TB in 80(3) 0 I(G) such that era BT g = gu g B = gv with u,v E Gk . So OT 8? 6T 8T (*) (g °’> B = (gu) B B gvu with v,u E Gk. = B . 6 5 aBTaTB With TQTB implies that 08 E §k(s). Since aTaBT E SC(s) fl I(G) .(*) then b) Sk(s) is normal in SO(S)° Let 9 be an element in S (s) and a in S (3). There exists a T in S (s) n I(G) O k 0’ _10 9 such that [Gk-l’aTa] S Gk° Let g E Gk-l' Then g E Gk-l since 9-1 E 80(3). We have 9-1 aT 9-1 (g ) a = g u (for some u E Gk) 9’1( )9 e e = g aTa = g u with u E G k. -1 -1 _ Since G aT 9 = 9 aeTe we have 6 Ice E S (s). a (y k The proof that Sk(s) is a normal subgroup of SO(s) is an adjustment of the above proof and we are done. Q.E.D. 40 From Proposition 3.1.7 it follows that given a chain of arbitrary subgroups of G s: G = C0 2 G1 2...2 Gt-l 2 Gt we get two normal descending chains of subgroups of 30(3) which are: 80(3) 2 S1(s) 2...2 St_1(s) 2 St(s) and 80(3) 2 §1(S) 2...2 St_1(s) 2 SC(s) In case s is a subnormal chain of subgroups of C, it is easy to see that an element 9 is in Sk(s) if and only if e induces the identity automorphism on the quotient groups Also an element 9 is in Sk(s) if and only if there eXists a Te(i) E 80(8) 0 I(G) depending on i such that G. 1 6Te(i) induces the identity automorphism on for G1+1 i = O,l,...,k-l. Definition 3.1.8. The series of subgroups SO(s) 2 81(8) 2...2 St-l(S) 2 St(s) is called the stability series of s for G. Similarly the series of subgroups is called the local stability series of s for G. At this point it might be worthwhile to give some examples to show that the local stability series of s is quite different from the stability series of s. 41 Example 1. (for a non solvable group G). Let ‘n 2 S and n # 6. Consider the symmetric group of degree n. Let G = G = S , G = An and G = <1>. We have 1 2 s: S 2 A 2 <1>. n n Then 80(8) = Sn and 81(5) = 82(3) = Sn (see Theorem K in the Appendix). But S1(s) = Sn. We claim that 82(3) = <1> for S2(s) is normal in SO(s) = Sn’ and the only non trivial normal sub- group of Sn is An (see Theorem L in the Appendix), which is not abelian. Example 2. (for a solvable group G). Take G = C0 = S3, G = A 1 3 and G 2 = <1>. We have the chain 3: 33 A_A3 g <1>. Then 80(3) = 81(3) = 82(3) = Sn (Theorem K in the Appendix) while 31(3) = S3, 52(3) = A3 since A3 is abelian and S3 is centerless. With respect to two different chains of subgroups of G we have the following result. ' ' . . . : = 2 2...2 = > PropQSition 3 l 9 Let 51 G CO > C1 > Gt—l 2 Gt <1 and 32: G = HO 2 H1 2...2 Ht-l 2 Ht = <1> be two chains of 9 arbitrary subgroups for G such that Gt for some 9 in A(G). Then e-1(Sk(sl))e = Sk(52) 42 and -1 ~ ~ 9 [Sk(sl)]e = Sk(82) for k = O,l,...,t. Proof: We only prove for the element Sk(s) of the local stability series. The proof may be adjusted to be valid for the stability series. -1 ~ . ~ a) Case k = 0. e (30(31))9 is a subgroup of 80(32) is clear. Also 9(§0(82))9-1 is a subgroup of 80(31). So -1 ~ _ ~ Now since I(G) is normal in A(G) we will also have e'1[§0 n I(G)]e = §O 0. We want to prove that e-l(§k(sl))9 is contained in Sk(82)° Let a be in Sk(sl). For 1 S i S k there exists Ta(i) E 80(3) 0 I(G) depending on i such that [Gi_1,aTa(i)] S Gi' Let h be an arbitrary element in Gi- 1. 9-1 Then (h) is in Gi-l' We have -1 . -1 (h)9 “Ta(l) = (h)9 u with u in Gi 9-1[aT (1)19 e 9 So (h) a = hu with u E Gi Therefore -1 . h-1(h)e [aTa(l)]9 3 U9 . Again e-1[aTa(i)]e = e-lae(Ta(i))e with (Ta(i))e e 30(s2) n I(G) by a). Since this is true for all i such that l s i s k 43 -1 ~ Similarly we can prove that e[Sk(sz)]e-1 is contained . ~ -1 ~ _ in Sk(sl)’ Thus 9 [Sk(sz)]e — Sk(sz). The last term St(s) of the stability series of a sub- normal chain has some nice properties as shown by A.D. Polimeni [IQ in Chapter III of his thesis. We will generalize some of his results to arbitrary chain of subgroups of G. We also show that §t(s), the last term of the local stability series, has also some nice properties. 3.2 Some Properties of St(s) and St(s) For a start, we recall here some results about SC(s) and St(s). Let s: G = C0 2 G1 2...2 Gt—l 2 Gt be a chain of arbitrary subgroups of G. Then Philip Hall theorem [7] says that the last term of the stability series St(s) is nilpotent of class at most (5). On the other hand Corollary 3.1.4 says that if St(s) is the last term of the local stability series of s, then St(S) ~ is solvable of derived length at most (t-l) + (g)' St(s) fl I(G) In other words St(s) is solvable modulo the inner automorphism group of G. We will use the following lemma to characterize the stability series and the local stability series of a chain 5 in a different manner. 44 Lemma 3.2.1. Let s: G = G 2 G 2...2 Gt 2 Gt = <1> be a O l -1 chain of arbitrary subgroups of G. Let N1 = CoreG (G1) be i-l the maximum.normal subgroup of Gi-l contained in 61' Then for any 9 in 80(3), N? = Ni for i = l,2,...,t. Proof: It is known that Core (Gi) = H CH Gi-l XEG, 1 i-l Let 9 be any element in 80(8). Then by definition So for x in Gi-l we have 9 9 e _ e x = x . — (Ci) (Ci) With x x 9 (Ci) E Gi-l° Thus 9 just permutes the elements of the conjugates class [Gi‘x E Gi-l} among themselves. So ( n 6’59: n G: xEGi_1 XEGi_1 i.e. N9=N.. Q.E.D. i i PropOSition 3.2.2. Let s = GO 2 G1 2...2 Gt-l 2 Gt = <1> be a chain of arbitrary subgroups of G. Let Ni = CoreG_ (Ci). 1. Define ~' :8. = e: . = 30(s) O(s) {9€A(G)\Gi Gi v1 0.1. .t} and recursively Sfi(s) = {e E Sk-1(S)\9\Gk_1 = <1>} for k = 1,2,...,t Nk 45 and also recursively Sfi(s) = [G E §é_1(s)\3 Te E 80(8) 0 I(G) such that Then we have II C U ...: U N U U ('f Sfi(s) Sk(s) for k and N. = ~ Sk(s) Sk(s) for k ll 0 v ...; V N V U ('7‘ M= a) 812(3) =Sk(s) for k=0,l,2,...,t. For k=0 it is true trivially. So suppose that S£(s) = Si(s) for any i < k. Let a be any element in Sk(s). Then a is in Sk_1(s). Since Sk-1(S) = S (s) by inductive hypothesis, I k-l . . ' a is also in Sk_1(s). On the other hand by definition of a, [Gk_1,a] is contained in Gk' We claim that [Gk—1’0] is also a normal subgroup of G For if g and y are tWO elements Of k-l' Gk-l then -1 -1 -1 -1 (g g0!)y = y a gay = [(gy>'1“] e [Gk_1,a] . But N = Core (G ) is the maximum normal subgroup of k Gk_1 k Gk-l contained in Gk' So [Gk-1’ a] is contained in Nk. Gk-l ‘ Since clearly a centralizes a fortiori a [Gk-I’d] ’ 46 Gk-l Nk Conversely let a be any centralizes i.e. . a! = g in Gk-l’ we have (gNk) gNk. That means g u an element of Nk' Therefore g-lga = u E Nk [Gk"1 )a] S Gk. So a is in Sk(s). ~' = ~ b) Sk(s) Sk(s) for k it is trivially true. So suppose Let a be an element in a is an element of Sfi(s). element in S£(s). Then for (1’ = gu with s Gk, i.e. = O,l,2,...,t. Again for k — O that §£(s) = Si(s) for i < k. Sk(s). There exists an T belonging to SO(s) fl I(G) such that [Gk-l’aTa] is contained in G As in part a) we then have k' Nk. This means, for g in Gk-l’ in Nk' CY'T Therefore (gNk) a = gNk "t Sk(x). Conversely if a is in S Ta 6 80(3) fl I(G) such that, for Q’Ta (gNk) : gNk’ OT So g a = gn with n an element _ O’Ta g g = n E Nk i.e a is in Sk(s). [Gk-l’aTa] is contained in O’T we have g a = gn with n i.e. a is an element of fi(s), there exists g in Gk-l of Nk. Hence 5 Gk Q.E.D 47 We will now examine the order of St(s) and St(s). Lemma 3.2.3. (A generalization of Theorem 3.1 of A.D. Polimeni [10].) Let s: G = GO 2 G1 2...2 Gt-l 2 Gt = <1> be a chain of arbitrary subgroups of a finite group G. If p is a prime divisor of \St(s)‘, then p divides \G‘. 12225.: Let 9 be any element of St(s) of order p, then 9 fixes Gt-l elementwise. Let r be the first r such that Gr is fixed elementwise by 9. Since 9 # l, r is greater such that x9 # x. P X(9 ) = than 0. There exists an x in Gr-l Then X6 = xg with g in Gr' It follows that x = xgp. Thus since Xe # x, g must be of order p. So p‘\G\. PrOposition 3.2.4. Let G be a finite group and s: G = G0 2 G1 2...2 Gt—l 2 Gt = <1> be a chain of arbitrary subgroups of G. If p is a prime dividing ‘St(s)‘, then p divides ‘G‘. 2229: Suppose there exists an element 9 in §t(s) such that 9p = l, but p does not divide ‘G‘. For k = 1,...,t there exist an depending on k, such that T 6 80(3) 0 I(G) x k This implies that Xk, e Xk and (gNk) = (gNk) for any g E Gk-l' ((x W _ k - (gNk) P = C CG(-fi-—-). In particular k xk centralizes gNk, i.e. (gNk) 9 = gNk and this is true for all k = l,2,...,t. So 9 is in fact an element of St(S)° Therefore p divides \St(s)\. By Lemma 3.2.3 p also divides ‘G‘ which is a contradiction. This contradiction proves the prOposition. Q.E.D. Corollary 3.2.5. If G is a finite p-group, p a prime, and s is a chain of arbitrary subgroups of G, then St(s) and St(s) are finite p-groups. Corollary 3.2.6. Let G be a finite p-group, p a prime, and : = 2 o o o = ' " let s G G0 G1 2 2 Gt-l 2 Gt <1> be a chain of arbitrary subgroups of G such that Gt-l is cyclic of order p. Then St(s) is a p-Sylow subgroup of S (s) and 9 E S _ t-l implies ‘9“‘St (s)|(p-1) . Proof: ~ Let 9 be any element of St-l(S) of p power order. 9 induces an automorphism on. Gt~JU.Since \Gt-l\ = p, A(Gt-l) is cyclic of order p-l (Theorem A in the Appendix). Thus for p-l (9 )= x. But (‘9‘,p-l) = l, we have > = <9> S.A(G). 80 x9 = x for all x E Gt-l’ and x E Gt-l’ x <91)-1 9 E St(s). So St(s) must be the p-Sylow subgroup of St 1(s). 49 Next, let 9 E §t-1(S)° As above ep-l induces the identity automorphism on Gt- Hence eP-l E §t(s) and by 1. the previous paragraph \e\“§t(s)‘(p-l). Q.E.D. Corollary 3.2.7. Let G be a finite Z-group and let s be a chain of arbitrary subgroups of G such that Gt-l is of order 2. Then §t_1(s) is solvable. Proof: By Corollary 3.2.6, St 1(s)~ has a normal 2-Sylow sub- - S (3) group St(s). The quotient group :£;1-- is then a finite St(s) §t_1(s) group of odd order. Therefore -—-—-——- is solvable S (s) by Feit and Thompson theorem. Thus St 1(s) is solvable, being an extension of a nilpotent group by a solvable group. If in Corollary 3.2.6 and Corollary 3.2.7 we replace St(s) and St_1(s) by St(s) and St-1(S) respectively, we get a similar result. Corollary 3.2.8. Let G be a finite p-group and s be a chain of arbitrary subgroups of G such that Gt-l is cyclic of order p. Then St(s) is a p-Sylow subgroup of St_1(s) and e 6 St-1(S) implies that ‘6“\St(s)\(p-l). Proof: Same proof as in Corollary 3.2.6. Corollary 3.2.9. Let G be a finite 2-group and let s be a chain of arbitrary subgroups of G such that Gt-l is of order 2. Then St_1(s) is a solvable group. Proof: Same proof as in Corollary 3.2.7. 50 So far we have examined St(s) and St(S) for an arbitrary chain of subgroups of C. Now if we restrict our attention to composition series or normal Hall chains (this term will be defined later) then we get more specific results. Proposition 3.2.10. Let G be a finite solvable group and s = G = GO 2 G1 2...2 Gt-l 2 Gt = be a composition series of G. Let p be the largest prime dividing ‘G‘, then the following two statements hold i) 30(3) is solvable ii) If the p-Sylow subgroup P of 80(8) is not <1> then. P is contained in St(s) and P is normal in 80(3). 11mg: Part i). Since G is a finite solvable group, G has a normal series 32 where each factor group is abelian. We can refine this series into a composition series Each S3. for G must be factor group of this composition series 53 cyclic of order r, r a prime dividing ‘G‘. The series 3 Gi—l and 33 are equivalent. So for i = l,2,...,t, ‘E—— is i cyclic of order a prime. Now the restriction of the action of Si 1(s) on is a homomorphism with kernel Si(s). S. (8) Hence —§:%§)— is isomorphic to a subgroup of the auto- i G._ morphism group of G This latter one is abelian because i G._ S._1(s) is cyclic. So 4—— is abelian for i = l,2,...,t. Gi Si(s) 51 S (5) Hence 0 is solvable. This implies S (s) is solvable, St(s) 0 being an extension of the nilpotent group St(s) by a solvable group. Part ii). Let 1 # e be any element of 80(3) of G. p-power order. 9 induces automorphism on each group é-l , i G,_ i = l,2,...,t. Since g is cyclic of order a prime r . i for example,AA( $-1) is cyclic of order r-l. But (p,r-l) = l, i because p is the largest prime dividing \G‘. So 9 induces G. the identity automorphism on each é-l for i = 1,...,t. 1 Hence 9 is in SC(S). Therefore P is contained in St(s). P is also a Sylow subgroup of St(S)' But St(s) is nilpotent. So P is a characteristic subgroup of St(s). Since St(S) is normal in 80(5), P is also normal in 30(3). Q.E.D. Definition 3.2.11. A subgroup H of a finite group G is called a Hall subgroup if ‘Hl and [GzH] are relatively prime. ' = = 2 ‘...“ ‘ = A normal chain 8 G N0 N1 2 > Nt-l 2 NC <1> where each N1 is normal in G is called a Hall chain if the Ni are normal Hall subgroups of G. t is called the length of the chain. Proposition 3.2.12. Let G be a finite group and s = G =‘N 2 N 2...2 Nt_ O 1 2 NC = <1> be a Hall chain. Then 1 §t(S) ITYGY— is nilpotent of class at most (t—l). Proof: Because 5 is a normal chain, I(G) is contained in §t(s). 52 S (s) -——————-—- ' '1 t t St(s)I(G) 13 n1 po en By Corollary 3.1.4 we know that of class at most t-l. To conclude the proof we observe that St(s) is con- tained in I(G) as the following theorem by R. Baer [1] shows. Theorem 3.2.13. Let G be a finite group and s = G =‘N0 2 N1 2...2 Nt-1 2 Nt = <1> be a Hall chain. Let Z(Ni) be the center of the subgroup Ni. Then St(s) is t nzmp isomorphic to W For the sake of completeness we will prove this theorem here. 2.29.9: Since Z(Ni) is characteristic in Ni and N1 is normal in G,ZCNi) is also normal in G. So pzmi) makes sense. t Clearly any element x in H Z(Ni) induces an auto- 1 morphism Tx belonging to St(s). So we want to prove that a) Case t = 2. Let a be an element of 82(3). 0 fixes N1 elementwise. By Lemma 1.2.1 0 induces the identity NG(N1) _ c 0’ ———-————— Sofor gEG,g =gc with CG(N1) CG(N1) automorphism on c in CG(N1). But 0 also centralizes S—u So c must be in 1 N1, i.e. c E CG(N1) fl N1 = Z(Nl). Now because N1 is a normal Hall subgroup of G, by Theorem 0 in the Appendix, there exists a subgroup K such that G = KNl' 53 Now consider the group KZ(N1) = M. M is fixed by or because for k E K, kc = kc with c E Z(Nl)° Z(Nl) is clearly a normal Hall subgroup of M. Using Theorem 0 of the Appendix . . . m again we know there eXISt an m in M such that K“ = K . But z m can be written as k z , with z in Z(N ). So K0 = K O. 0 O O 1 Consider a7 _1. We have already k“ = kc with c e Z(Nl). 0 GT 1 20' T also belongs to S (s). So k = kc'c with c' 20-1 2 II in Z(Nl)' But by above kc'c E K, so c'c identity. This CYTZO_1 implies k = k. QT 20-1 is then the identity automorphism on G. So a = T Case t > 2. Suppose the theorem is true for any group with a chain of length < t. Let a be an element in St(s). a induces an auto- morphism on N1. Since N1 2 N2 2...2 Nt-l 2 Nt = <1> is a Hall chain of N1 with length t-l, there exists an element n1 in t n Z(Nl) such that aTn induces the identity automorphism on 2 1 N1. Also clearly aTn stabilizes the chain G Qle A 1. By the 1 first case there exist n E Z(N ) such that am T = identity 2 1 n1 n2 on G. So a = «r‘ln. Q.E.D. n1 2 Proposition 3.2.14. Let G be a finite group. Let s = G = GO 2 G1 2...2 Then G is nilpotent if and only if St(s) = St(s). Gt 1 2 Gt = <1> be a chief series for G. Proof: Assume G is nilpotent. Its ascending central series terminates in G. We can refine the ascending series into a 54 chief series and all chief factors of this series will be central. By the Jordan-Holder theorem it then follows that the chief Gi Gi+1 ~ implies that St(s) = SC(s). factors are centralized by the element of G. This clearly Conversely if St(s) = St(s), then St(s) is nilpotent. Since I(G) is a subgroup of St(s), I(G) is nilpotent. So G is nilpotent. Q.E.D. BIBLIOGRAPHY 10. 11. 12. 13. BIBLIOGRAPHY R. Baer, Die Zerlegung der Automorphismen€3ruppe einer endlichen Gruppe durch eine Halliche Kette. Journal fur die Reine and Angewandte Mathematik, 220 (1965) 45-62. E.C. Dade, Locally Trivial Outer Automorphisms of Finite Groups. Mathematische Zeitschrift, 114 (1970) 173-179. , Automorphismes Exterieurs Centralisant tout Sous groupe de Sylow. Mathematische Zeitschrift, 117 (1970) 35-40. _ G. Glauberman, On the Automorphism Group of a Finite Group Having No Non-Identity Normal Subgroups of Odd Order. Mathematische Zeitschrift 93 (1966) 154-160. D. Gorenstein, Finite Groups (New York, Evanston, and London: Harper and Row, Publishers, 1968). M. Hall, The Theory of Grogps (New York: The Macmillan Company, 1968). P. Hall, Nilpotent Groups, Canadian Mathematical Congress, Summer Seminar, University of Alberta, 1957. H. Liebeck, The Automorphism Group of Finite p-groups. Journal of Algebra, 4 (1966) 426-432. D.S. Passman, Permutation Groups (New York , Amsterdam: W.A. Benjamin, Inc., 1968). A.D. Polimeni, A Study of the Automorphisms and Chains s of a Finite Group. Ph.D. Thesis at Michigan State University, 1965. C.H. Sah, Automorphisms of Finite Groups. Journal of Algebra 10 (1968) 47-68. W.R. Scott, Group Theory (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1964). I{.Wielandtand B. Huppert, Arithmetical and Normal Structure of Finite Groups. 1960 Institute on Finite Groups, Proceeding of Symposia in Pure Mathematics 6 (1960) 17-38. 55 14. 15. 56 J.R. Weaver Solvable Automorphism Groups and an Upper Bound for A(G)‘. Proceedings of the American Mathematical Society 27 (1971) 229-235. D.L. Winter, The Automorphism Group of an Extraspecial p-Group (unpublished). APPENDIX APPENDIX I. THEOREMS. Theorem A [12]. The automorphism group of a cyclic n-l p-group,2$f>a prime, is cyclic of order o(pn) = p (p-l), pn is the order of the group, 6 is the Euler 6 function. Theorem B [6]. Every group of order pmqn, p and q primes, is solvable. Theorem C [12]. If G is a finite solvable group then CG(Fit(G)) C Fit(G) where Fit(G) is the Fitting sub- group of G. Theorem D [2]. If G is a finite group, then the group LT(G) of all locally trivial outer automorphisms of G is a normal nilpotent subgroup of Out(G). Theorem E [3]. If G is a finite group, then the group SC(G) of all outer automorphisms centralizing all Sylow sub- groups of G is a normal solvable subgroup of Out(G). Theorem F [5]. The special linear group SL(2,q) of 2 X 2 matrices over a field of q = pn elements, p = an odd prime, has the following properties. i) SL(2,q) contains cyclic subgroups of order q-l and q+l ii) A 2-Sylow subgroup of SL(2,q) is generalized quaternion. 57 58 Theorem G [9]. Let G be a generalized quaternion group then the following is true i) if ‘G‘ = 8 then A(G) is isomorphic to the symmetric group of degree 4. ii) if \G\ 2 16 then A(G) is a 2-group. Theorem H [12]. 1) GL(2,2) = SL(2,2) z PSL(2,2) 2 S3 2) PSL(2,3) 2 A1t(4) (S = Symmetric group of degree 3) 3 (A = Alternative group of degree 4). 4 Theorem I [8]. Let N be a characteristic subgroup of G. Let 3!: A1(G;N) be the group of all automorphisms of G which fix %' elementwise. Then MI is normal in G and fig?)- is isomorphic to the subgroup of automorphisms in A(S) that can be extended to automorphisms of G. Theorem J [15]. Let p be an odd prime and let P be an extra-special p-group of order p3 and exponent p . Let I be the group of inner automorphisms and let H be the normal subgroup A(P) consisting of all elements of A(P) which act triviallyon Z(P). Then A(P) = H where 9 has order p-l, H H = <1> and H/I has order p. Theorem K [12]. If n 2 5 the alternating group of degree n is simple. Theorem L [12]. If 02.3.,nf6 the automorphism group of Sn is isomorphic to Sn’ Sn being the symmetric group of degree n. 59 Theorem.M. If n # 4, the only non-identity proper normal subgroup of Sn is An' Proof: ‘Sl| = l, ‘52] = 2. The theorem is true trivially. ‘83] = 6. So a prOper subgroup N of S3 must have order 2 or 3. If it has order 2, then it contains, for example, the cycle (12). If N is normal then (13)(12)(13) = (32) 4 N. Contradiction. So N must have order 3. N must contain all 3 cycles, i.e. N = A3. Now for n 2 5. Let N be a prOper normal subgroup of S . N fl.A is a normal subgroup of A . So N PIA = <1> or n n n n N HAn = An since An is simple. If N (\An = <1>, then S \r n If c #a,b, then (ac)(ab)(ac) = (cb) é N. Contradiction. 80 ‘N‘ = 2 since n‘ = 2- 7 so a 2-cyc1e for example (a b) E N. N (\An = An,‘which implies N = An because N is prOper. Theorem N [7]. (Philip Hall Theorem). Let G = G 2 G 2...2 G 0 1 t-l 2 Gt = <1> be a chain of subgroups of G and let A be a group of automorphisms of G such that [Gi_1,A] s Gi for each i = l,2,...,t. Then A is nilpotent of class at most (;)' Theorem 0 [5]. (Schur-Zassenhaus-Feit-Thompson). Let H be a normal Sn-subgroup of G. Then we have i) G possesses an Sn,-subgroup K which is a comple- ment to H in G. ii) Any two Sn,-subgroups of G are conjugate in G. 60 Theorem P [6]. If P is a p—group of order p“, 6 the intersection of the maximal subgroups of P, and [Pzé] = pr, then the order of A(P), the group of auto- r(n-r)e r morphisms of P, divides p (p ). The order of A1(P), . . . P . . . . the group of automorphisms fix1ng g' elementWise, is a diVisor of r(n-r). illllWIHI1||lHINW||111WllmmHIIHIHIIHIIIHIH