ll w M \ fl \ilIHWWW l 1 *WIIW AUTOMGRPWSMS 0F FQNETE p'GROUPS W! L \ \ SEE: W W‘s fa? the Degree of Ph. D. E‘efiflfiiGM-é STATE UE‘éEVERSi’W CLEVE H. RES 1969 "f -‘Dl- LIBRARY Michigan State University This is to certify that the thesis entitled Autamorphisms of Finite p - Groups presented by Clive M. Reis has been accepted towards fulfillment of the requirements for _£h‘D__ degree inJiflthgtiea 02.7%. Major prflssor Date Feb. 5; 1969 0-169 ‘1 ' amaloiu " " 3 “MG & SUNS' "‘ " 800K BIND!" m0. ‘ ABSTRACT AUTOMORPHISMS OF FINITE p-GROUPS By Clive M. Reis In the second chapter certain subgroups of the automorphism group of a finite p-group are investigated with a view to determining upper bounds for the derived length and exponent of the subgroups. In the third chapter, the full automorphism group of a finite p-group of maximal class is investigated and bounds for the derived length and order of the group are obtained. Many of the subgroups of the automorphism group that have been studied are associated in some way with subgroups of G. In chapter II of this thesis, we first investigate those subgroups of the automorphism group of G inducing the identity on certain abelian subgroups of G. The final result is the following Theorem If A is maximal subject to being abelian and of exponent +1 s-t p(c ) Pt, then lEAut(G|A)] S log2 + c and eprut(G|A) s who where expG = pS and ch = c. The proof of this result can be divided into 5 distinct steps; each succeeding step is dependent for its proof on the preceding one. lst Step We obtain information about Aut(G|Qt) for regular p-groups, p odd and for Z-groups of class not greater than 2. 2nd Step We investigate Aut(G|A) where A is maximal subject to being normal, abelian and of exponent pt. Again we assume G is regular if p is odd and of class at most 2 if G is a 2-group. Clive M. Reis 3rd Step Aut(GIOt) is investigated, no restriction on the group G. 4th Step We investigate Aut(GlA), A still maximal subject to being normal, abelian and of exponent pt. 5th Step We no longer require A to be normal and obtain the quoted theorem. Finally, as a generalization of the preceding, we obtain in- formation about those automorphism subgroups inducing power auto- morphisms on certain abelian subgroups of G. In chapter III we prove the following Theorem If G is a p-group of maximal class, IGI = pm, m > 3, then (i) xEAut(c)] s logz 8(m-l) (ii) Aut(G) = P.C where P is the p-sylow subgroup of Aut(G). Furthermore ptn 3 IP' S pZm-3 and |C||(p-l)2. Intermediate steps in proving this theorem are as follows: lst Step Aut(G;I‘2) s Aut (G > F2 >. . .> I‘m th 1 term of the lower central series of G, G any nilpotent group. > 1) where F, is the -l 1 2 2nd Step If E is an elementary abelian group of order p , generated by P and Q, then the group A of automorphisms given by P a P1 i 1:-°'3(p-1) 02°°'3(p'1) QrQPL *« is of order p(p-l) and of derived length 2. 3rd Step The quoted theorem follows easily from steps 1 and 2. AUTOMORPHISMS OF FINITE p-GROUPS By .li.‘§' ’ Clive M.‘ Reis A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1969 To Sandy, Helen, Rhean and Jonathan ii ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Professor Ti Yen for his help in the preparation of this thesis. He gave unstintingly of his time to discuss the many problems which arose and contributed many helpful suggestions. iii TABLE OF CONTENTS Page INTRODUCTION .................................................... 1 CHAPTER I . . . . .................... . .............................. 3 CHAPTER II ................ . . . . . ................................. 7 CHAPTER III .............. ................................. 32 INDEX OF NOTATION ............................................... 41 BIBLIOGRAPHY ..... . ...... . . . . . ................................... 42 iv INTRODUCTION The automorphisms of a group may be studied from two distinct points of view. 0n the one hand we may postulate the existence of an automorphism or a group of automorphisms having certain properties and we may ask if we can determine to some extent the structure of a group G possessing such automorphisms. 0n the other hand, we may impose restrictions on the group G and we may then investigate the structure of Aut(G) or of some of its subgroups. Knowledge of the structure of Aut(G) may then be used to obtain further information of the structure of C. It is from this latter point of view that we shall investigate certain subgroups of the automorphism group of a finite p-group in Chapter II. In Chapter III, information about the full automorphism group of a p-group of maximal class will be obtained. Most of the subgroups of Aut(G) that have been studied are those related in some way to a subgroup or chain of subgroups in G. Thus, for example, P. Hall has proved that the group of automorphisms stabilizing a chain of subgroups in G, ending in the identity, is nilpotent. Another example is a result of N. Blackburn which states that the automorphisms of a finite p-group, fixing elementwise a subgroup which is maximal subject to being abelian and of exponent pt, is a p-group. In this investigation, we shall be primarily concerned with those subgroups of Aut(G) inducing the identity, or more generally, 1 inducing a power automorphism on certain subgroups of G. We shall obtain an upper bound for the derived length and for the exponent of the automorphism groups under investigation. In some cases, we shall even be able to obtain bounds for the order of the automorphism group. CHAPTER I Preliminary Results In this chapter we introduce some of the concepts to be used in subsequent chapters. We shall also prove a number of theorems which form part of the folk-lore of the subject of automorphisms of groups. Section 1 Definition 1.1.1 Let G be a group and let xi be elements . . -1 -l of G. We write the expression x1 x2 xlx2 as [x1,x2] and call it a simple commutator of weight 2. Define a simple commutator of weight n inductively by [x1:'°-9xn_13xn] = [[x13°'°:xn_1]:xn]- I“ =< > 22. ' Let k(G) [x1, ’Xkllxi E G , k The series G E T2(G) E...§ Fk(G) E.... is called the lower central series of G. Clearly each Pi is fully invariant and it may be proved that [F.(G),G] = <[x y]|x E F,(G) and y E G> = F, (G) for i = 1,... 1 1 1+1 provided F1(G) is interpreted as G. Definition 1.1.2 If Fc+1(G) = l and FC(G) # l, G is said to be nilpotent of class c. Theorem 1.1.1 (P. Hall)[12] If G is a nilpotent group, S ' ' = [F1,FJ] _ Fi+j for 1,3 2,3,. The following commutator relations will be used: Lemma 1.1.2 If a,b,c E G, G any group, then 3 4 [a.c][a.bl[a.b.cl [a.cl[a.c.bJ[b.c]. (i) [a.bc] (ii) [ab,c] Proofs of these relations will be found in M. Hall [11]. The following theorem lists the elementary results we shall need about p-groups. Theorem 1.1.3 Let G be a finite p-group. Then (i) G is nilpotent (ii) If H i G, NG(H) ¥ H. (iii) @(G), the Frattini subgroup of G is the inter- section of all maximal subgroup of G. P is characterized by the property that it is the smallest subgroup H for which G/H is elementary abelian. As a general reference for the theory of finite p-groups we give P. Hall's paper [12]. Section 2 In this section we list and prove some of the simpler theorems on automorphisms. Notation (i) Aut(G) denotes the full group of automorphisms of G. (ii) expG is the least common multiple of the orders of the elements of G. Definition 1.1.3 Let H E G. Then if a E Aut(G) and x-lxa E H for all x E G, a is said to be an H-automorphism. The set of all such automorphisms is denoted by Aut(G;H). This set is actually a group, and when Hid G, it is just the group of auto- morphisms of G inducing the identity automorphism on G/H. We sometimes say that a stabilizes G/H. (b) Let G = G > G >...> Gm be a chain of subgroups. O 1 Then the set of automorphisms 0 having the property that for all x 6 G1, (i = 0,...,m-l), x-lxa E G will be denoted by i+l’ Aut(G > G 1 >...> Gm). This set again forms a group and we shall call this group the group of automorphisms stabilizing the chain G > G >...> G . l m (c) Let H é G. Then Aut(G‘H) will denote the group of automorphisms fixing H elementwise. We note that if Gm = 1 in > ..> (b) above, Aut(G > C1 . Gm_1 Lemma 1.2.1 Let G be a finite p-group and let > 1) a Aut(clcm_1). G = G0 > C1 >...> Gm 1 > 1 be a chain of subgroups. Then Aut(G > G >...> G 1 m-l > 1) IS a p-group. Proof By induction on the length of the chain. If m = 2, and a E Aut(G > C1 n xa = xgg. Hence, if n = expG, xa = x for all x E G. We note > 1), then xa = xg1,g1 € G1. Therefore n that we have incidentally shown that eprut(G > C1 > 1) § expGl. Assume inductively that the result is true if the length of the chain is less than m and let a E Aut(G > G >...> G > 1). Thus the l m-l restriction of a to G1 is in Aut(G1 >...> l) and by induction k R up fixes G1 elementwise for some k. Therefore up 6 Aut(G > G1 > 1) and by what we have shown above, a is of p-power order. By induction, we are done. Lemma 1.2.2 Let G be a p-group, Hmd G and suppose CG(H) é H. Then Aut(GIH) = Aut(G > H > 1) and is an abelian p-group of exponent not greater than epo. 2599; Let g E G and h E H. Then since H 4 G, [g,h-1] E H. Let a E Aut(GlH). Then [g,h-1]a = [g,h-1] = [ga,h-1]. Therefore -1 l -a a -l a -l g h g h- = g h g h . Hence g g h = hg‘wgm1 for all h E H. Thus gag-1 6 CG(H) E H for all g E G and the result follows. By the previous lemma Aut(GIH) is a p-group. That it is abelian is proved as follows: Let 01,3 6 Aut(GlH). Then x0,8 = (xh )8 = (xh )h = xh h = xsa since, from what a B a a B we proved above, ha and hB E Z(H). We shall have occasion to use a generalization of the above. Theorem 1.2.3 Let G > G >...> G = 1 be a series of normal subgroups. Then clAut(G > G >...> 1) § m-l. A proof of this will be found in [13]. CHAPTER II Section 1 In this section we prove some theorems about certain subgroups of the automorphism group of two specific classes of p-groups. 1n the case p is odd, the results will be proved for regular p-groups and in the case p even for groups of nilpotent class at most 2. When p is odd, the regular groups form a more comprehensive class of groups than that formed by the groups of nilpotent class at most 2. When p = 2, however, the latter class is bigger since regular 2-groups are abelian [14]. Later on, when we extend these results to arbitrary p-groups, we shall need the theorems to hold for groups of nilpotent class at most 2. Thus in order to obtain the maximum degree of generality, we are led to consider the 2 cases separately. Definition 11.1.1 A p-group is said to be regular if, given any positive integer n and any pair of elements a and b of C, it is always possible to find elements c1,...,cr all belonging to n n n n n ' and satisfying the equation (ab)p = ap bp c: ....c: . P. Hall [12] proves the following Theorem 11.1.1 If a and b are any two elements of the pn P“ -1 pn regular p-group G and if a = b , then (ab ) = l, and con- versely. 1n the same paper, P. Hall proves that the order of the product of two elements of a regular p-group is no greater than the maximum of the orders of the factors. This means that Qt = in a regular p-group. It is on theorem 1.1 that the basic lemma for the case p odd will depend. For the case p = 2, we shall depend on the following Lemma 11.1.2. [11] If 'G is a group of class not greater than 2, then for all a,b E G and for all integers n, n n (2) n (2) c“) (ab)n = anbn [b,a] 2 = anbn[b ,a] = anb [b,a ]. The following lemma is the cornerstone of many of the proofs: Lemma 11.1.3 (a) Let G be a regular p-group (p odd) and let. exp G = p8. Then Aut(GIOt) stabilizes G/Qs-t .(b) Let G be a 2-group generated by x1,...,xn where Ixil E 28 for all i. Let G be of class at most 2. Then Aut (clot) a Aut (c; s-t+l Proof (a) Let a E Aut(GIQt) and let x E G. Then S-t S-t s-t S-t p | E pt. Therefore xp 6 at and thus (KP )a = xp t S't S-t - 3. Therefore (xa)p = xp . By theorem 11.1.1, (xax 1)p = l and a -1 hence x x E ns-t' (b) Let a E Aut(G|fl ) and let xi 6 {x1,...,xn}. 2s-t+1 t-l s-t+ Then Ixi § 2 and xi 6 0t as above. Therefore 2s-t+l -l a 2s-t+l _zs-t+l 2s-t+l a _1 ( 2 ) (xi xi) = Xi Xi [xi’xi ] s-t s-t+l = [xi ,XiIJ(2 -1) = 1' Therefore x x: E O i s-t+l° r Now let x E G. Then x = H y, where y, E {x ,...,x }. When i=1 i i 1 n r = 1, x-lxa E 0 <0 > as we have just seen. é s-t+l s-t+l Assume inductively that when x is the product of at most _1 Q n+1 r factors, x x 6 ' Suppose now x = iHlyi -l a _ -1 -1 -l a a a d Then x x - xr+lyk ...y1 y1Y2---Yryr+1 - -1 ‘1 a a whe e = H " yr+1y y yr+1 r y . yi i=1 . . -1 a By induction, y y E ' Therefore -1 a. -1 a -a -1 d’d = O O O >. x yr+l yr+1 (yr-+1 y y yr+l) Whmh 13 1“ <0s-t+1 Remark (b) cannot be improved to give the same result as in '(a) as can be seen by considering the quaternion group. In this case 01 is the center of the group and expG = 22. But Aut(GIOl) = Aut(G) : S the symmetric group on four letters. However, 4, Aut(GIfll) cannot stabilize G/O2 1 = 6/01, for if it did, Aut(G) would stabilize G > 01 > 1 and hence would be abelian by lemma 2.2 in chapter 1. Theorem 11.1.4 (a) Let G be a regular p-group (p odd) and let expG = ps. Then S-t g > > . > = _ Aut(GIOt) Aut(G as"t .. > ns_kt 1) where k { t and {x} denotes the first integer greater than or equal to x. S-t Hence, by the theorem of P. Hall 1.2.3, cl[Aut(G|0t)] E {—E—}. Furthermore Aut(G‘Ot) is a p-group. (b) Let G be a 2-group of class at most 2. Let G = where Ixil é ZS. Then if t > 1, 1,.. Aut(GlOt) .5. Aut(G > > 1) >>...> t+l Qs-t >'...>'QS_kt > 1. But k is the first integer for WhiCh S'kt g t or, solving the inequality for k, the first integer for which k.§ EEE. Therefore k = {iii (b) Since <98 is generated by those elements -r(t--1)> s-r(t-l) of G of order not greater than 2 , lemma 11.1.3(b) is applicable and shows that Aut(Glflt) stabilizes <5} /<0 s-r(t-l)> s-(r+l)(t-l)> :>,> 1) ' g > > >... Therefore Aut(GlOt) Aut(G' .[Aut(GIOt)] i logzs-l/t-l. Corollary 11.1.5 with a slightly coarser bound can be proved by using lemma 11.1.3 and we shall give the proof in the following. The method of prood has the merit of making the structure of Aut(Giflt) more transparent. Theorem 11.1.6 (a) Let G be a regular p-group, (p odd), expG = ps. Then Aut(GIOt) e Aut(GIOzt) ewe Aut(GIOzkt) a 1 is a normal series with abelian factor groups. k is the first integer for which 2kt E'%. Thus lEAut(GIOt)] é {logZS/t} (b) Let G be a 2-group of class not greater than two. Then if t > 1, > 2 E”; > E 1 Aut(GIdlt ) Aut(G| 2t-l is a normal series with abelian factor groups. k is the first integer for which Zkt - 2k + l g.§§l. Thus 1[Auc(c|)] a {logzs-jl/t-l}. Proof (a) Consider LL: Aut(GIOZr ) -*Aut(02 where u. t is the restriction mapping. Then ker u = Aut(GIQ2r+1 ) and t Im u E Autdl But Aut(02r+1t|02rt) stabilizes 2r+ltinzrt)' 12 021""1t > ert > 1° Therefore Aut(GlQ2r )/Aut(G|02r+1 ) is abelian and (a) is t t proved. (b) is proved similarly, except that the restriction map must be taken from Aut(G|<fl >) to Aut(<02r+1 21«1.1-H» t- 2rt-2r+l Theorem 11.1.7 (a) Let G be a regular p-group (p odd), expG = p8. Then eprut(G|0t) .5. pS-t. (b) Let G be a 2-group of class at most 2, 3(s-t) s G = where Ixil § 2 . Then eprut(GIQt) E 2 for 1,.. t > 1. Proof (a) We shall show that if a E Aut(GiOt), R up 6 Aut(cln for all k a 1. t+k) Let 1).: Aut(GIO ' be the restriction mapping. t+k- 1) "' Antmwk) Then ker uni Aut (GIG and 1m 11 .5. Aut (O 0 ). But by lemma t+k| t+k-1 t+k|nt+k-1) Stabilizes ot+kmt+k--(t+k-1) = Oak/O \But 01§0t+k_1 for k-E 1. Therefore Aut(G IO nt+k > 01 > 1 and Aut(GIO Hence if a e Aut(GIOt), a? e madam) and eprut(G|Qt) a p t (b) (i) We show first that exp<flt> E 2 +1. Let t+k) 11.1.3(a), Aut(G 1. t+k-l) stabilizes )/Aut(GIOt+k) is elementary abelian. t+k t+k-l S-t r - t x€<fl>. Then x= Hx,|x.|§2t. If r=l,|x|§2. Assume t ' i=1 i i . . ' t+1 inductively that if x is the product of at most r xi'S, IXI ‘ 2 . r+1 Let x = H x. . 1 i=1 t+1 2t+1 Then x = (x1,...,xr,xr+1) 2t+1 2t+1 2t+1 (2 ) = (X1,...,Xr) Xr+1 [Xr+1, xls'°':x] t t+1 _ 2 (2 -1) . . - [Xr+1, X1,-~-,Xr] by induction = l l3 .. . _, > (11) Let u. Aut(GIOt) Aut(<flt+1) where p. is the restriction. ker u. = Aut(G|<flt+1> By lemma 11.1.3(b) Aut(dlt+1>|<flt>) stabilizes <11 5 ) and 1m p. _ Aut (<11t +1>|). > >, ' t+1 /<fl2 Since E > E . > > ' ' t 2, <02 Hence Aut( > <02> > 1. But exp<02> E 2 by (i) and hence Aut(G|) 3 5 . . . ex Aut (914%,?) _ 2 . A Simple induction then shows that eprut (c|) a 2 3(s-t) We next ask: Under what circumstances does Aut(GIOt) strictly contain Aut (GIG )? We can only answer this when t is "big t+1 enough" for purely non-abelian groups. Definition 11.1.3 A non-abelian p-group G is said to be purely non-abelian if it has no abelian direct factor. Theorem 11.1.8 (a) Let G be a regular purely non-abelian p-group (p odd), expG = ps. Let expG' = pn and epo(G) = p2 where n < s. Then Aut(GIQ ) é Aut(Glflt) if t E max(n,s-z) and t < s. t+1 (b) Let G be a purely non-abelian 2-group, S ch S 2 .and G = , lin § 2 . Let expG' a 2n and let 1,.. epo(G) = pz. Then Aut(G|<0H_1>) é Aut(G|<flt>) if t E max(n,s-z) and <flt> 95G. 11192;. (a) ExpG/Ot = ps-t é pz. There is therefore a homo- morphism 11: G/Ot -' Z(G) which is a monomorphism when restricted to'a cyclic factor of G/Ot. There is therefore an element xflt E G/Ot, Ixfltl = p, such that 110de 3‘ 1. Let B: G -+ G/Ot be the natural homomorphism and let P == 118. Then 9 is ahomomorphism from G to Z(G) which annihilates at but does not annihilate Ot+1. 14 Consider the mapping a: G v G given by a: x d xp(x). This is clearly always a homomorphism. Moreover, it has been shown [1] that when G is purely non-abelian, a is an automorphism. Clearly or E Aut (GIOt) but or Q Aut(GIOt+1) . (b) The proof is similar to that given in (a) except that in this case exp<flt> é 2t+1 and the element x whose order mod is 2 and which is not in the kernel of u: G/<flt> -' Z(G), t+2 . . has order not greater than 2 . (We cannot say it has order .§ 2t+l.) Corollary 11.1.9 (a) If G is a purely non-abelian regular p-group (p odd) such that expG' < expG and Qt ¥ G, then Aut(GiOt) # l. (b) If G is a purely non-abelian 2-group of class 2 such that expG' < expG and. <flt> # G, then Aut(G|<flt>) aé 1. P£22£_ Apply preceding theorem. Corollary_11.l.10 Let G be a regular purely non-abelian S-l p-group (p odd) with epo(G) 2 p and expG' = p. Then > Z . Aut(GIOt) e Aut(G|0t+1) for all s c _ 1 Proof max(n,s-z) = 1. Hence result. Finally we obtain a rough lower bound for |Aut(G|
)l when t is big enough. We first prove the following Lemma 11.1.11 Let A be an abelian p-group, A = C(p )®...® C(p ) and let B = C(pk) where k 2 avi for all i. Let a = Edi. Then there are at least pa distinct homomorphisms of A into B [8]. 15 a 0. Proof Clearly there are p t homomorphisms of C(p 1) into a. B. There are therefore at least H p 1 = pa distinct homomorphisms i of A into B. Theorem 11.1.12 (a) Let G be a regular purely non-abelian p-group (p odd), expG = ps, expG' = p and expz(G) = pz. Then if t E max(n,s-z), |Aut(GlOt)| E IG: 0 I (b) Let G be a purely non-abelian 2-group of class 2, G = )| E IG: <flt>| £322; (a) ExpG/Ot é epo(G). Thus by the previous lemma there are at least IG: Qtl different homomorphisms of G/Ot into Z(G), say u1,...,uk where k = IG: Otl. Then oi: x qxuiB(x) (where B is the natural homomorphism from G to G/Ot) are distinct automorphisms in Aut(G‘Ot). (b) is proved similarly. We shall now obtain bounds for the derived length of Aut(GIA) where A is maximal subject to being normal, abelian and of exponent t p . Theorem 11.1.13 (a) Let G be a regular p-group, (p odd), expG = ps. Let A be maximal subject to being normal abelian and t of exponent p . Then >.[Aut(c|A)] a logZZS/t and s P IIA eprut (CIA) (b) Let G be a 2-group of class at most 2. Let G = where |xiJ é ZS for all i. Then if A is 1" maximal subject to being normal abelian and of exponent pt, t > 1, l6 1[Aur(c|A)] .5. log2'2(s-l/t-l) and 243-3t+l IIA eprut (GIA) Proof (a) Let u: Aut(GIA) -* Aut(flt) be the restriction mapping. Then keru = Aut (GIQt), 11m. .5. Aut(QtlA). But by Alperin [2.], C0 (A) E A, and hence by lemma 1.2.2, Aut(OtIA) is abelian. t By Corollary 11.1.5(a), 1[Aut(GIOt) é logzs/t. Therefore 1[Aut(c|A)] a logZS/t + 1. Furthermore, by theorem 11.1.7(a), eprut(G|Ot) é ps-t and by lemma 1.2.2, eprut(OtIA) é pt. Therefore eprut(G|A) é p8. (b) Let u: Aut(GIA) ~>Aut(CG(A)‘A) be the re- striction map. Since CGQA) E.A, CG(CGQA)) é CGCA). Therefore keru = Aut(G'CGCA)) is abelian by lemma 1.2.2 and Imu é Aut(CGCA)lA). But by a result of Alperin [2], G0 (A) é A. Therefore Ot[CGCA)] = A. Therefore x[1mu] é log2(s-l/t-l) and hence 1EAut(GIA)] é log2(s-l/t-1) + 1. Furthermore, since 3 - 2 (S t) by theorem 11.1.7(b) and 24s-3t+l. Imp é Aut(CGCA)|A), eprmu é expkeru E 28+1. Therefore eprut(G|A) § In a regular p-group (p odd) there is an interesting connection between the exponent of A, (where A is maximal subject to being normal, abelian of exponent p) the exponent of G and the class of CGQA) as can be seen in the following: A E Z(CG(A)). Thus CG(A)/Z(CG(A)) GAut(cG(A)|A). But as we saw earlier, clAut(CG(A)IA) < s/t, since A = Ot(CG(A)). Therefore chGCA) <’s+t/t. Thus for example, if t E 5/2, CG(A) is of class at most 2. For the next few theorems we shall restrict our considerations to regular p-groups, p odd or even. Of course, when p is even we 17 are merely dealing with abelian groups. Similar results cannot be proved by the same methods for 2- groups of class 2 because in general in such groups exp(<flt>) é pt+1 whereas we need exp(<flt>) E pt as will be seen. Notation. Ck. will denote the set of elements which are pt-th powers of elements of G. P. Hall [12] has shown that in a regular p-group this set actually forms a group. Theorem 11.1.14 If G is a regular p-group, p-odd or even, then Aut(G;Ot) = Aut(GIUt). . t t t Proof Let Q! E Aut(G‘Ot). Then xp = (xp )0, = (xQ’)p for -l a pt all x 6 G. Therefore, by theorem 11.1.1, (x x ) = 1. Hence a E Aut(G;Qt). Conversely, suppose a E Aut(G;Ot). By the converse _ t (x lxa p = 1 t t t of the same theorem, since p = (x0)? = (xp )0 , x and hence or E Aut (Glut) . Therefore Aut (G;Ot) = Aut(GlUt). Corollary 11.1.15 Let G be an abelian p-group, p odd or even, which has no direct factor which is cyclic of prime order. Then Aut(G;01) = Aut (GlUl) = Aut(G > 01 > 1) i.e. Aut (G;01) is elemen- tary abelian. Proof Since G does not have a cyclic group of prime order a a as direct factor, G = C(p 1)‘®..J8 C(p n) where oi > 1. Therefore P 0’1"]- dn'l 01 = G : C(p ) ®...® C(p ). Hence ()1 2 01. By the pre- ceding theorem,.Aut(G;fll) = Aut(GlUl) é Aut(GIfll). Therefore Aut(G;01) = Aut(G >'01 > 1) and the theorem is proved. Corollary 11.1.16 Let G be a p-group with 01 § Z(G) and such that Z(G) has no direct factor which is a cyclic group of 2 order p. Then 1[Aut(o;nl)]§ 2 and eprut(G;01)-‘- p . 18 Proof Let u: Aut(G;01) ~.Aut(z) where u is the restriction map. Then. Inn 5 Aut(Z;fll(Z)) and by corollary 11.1.16, the latter group is elementary abelian. Furthermore, keru = Aut(Ggfll) n Aut(G|Z) é Aut(G >01 > 1) since 01 § 2. HA Therefore 1[Aut(G;01)] 2 and I 'O eprut(G;01) 5 Section 2 In this section we extend many of the results of section 1 to arbitrary groups. In general the bounds obtained in this section will be coarser. The transition from the special classes of groups we have been considering so far is effected by the use of the Feit-Thompson [7] "critical subgroup" of a p-group. This subgroup, which we shall henceforth denote by K, has the following properties: (i) K is characteristic of class at most 2. (ii) CG(K) é K and K/Z(K) is elementary abelian. (111) [G,K] a Z(K). Theorem 11.2.1 (a) Let G be a p-group (p odd), expG = p8. Then lEAut(G|Ot)] é log223/t and eprut(GIOt) § pzs-t. (b) Let G be a 2-group, expG = ZS. Then 24$-3t where lEAut(G|Qt)] 2 log22(s-l/t-1) and eprut(G|0t) é t > 1. nggf (a) Let K be the Feit-Thompson critical subgroup of G and consider u: Aut(GIOt) ~.Aut(K) where u is the restriction map. keru. = Aut(GI<flt>'K) = Aut(G > <9t>'1( > 1) by letnma 1.2.2 since CG(<flt>°K) .5. K. Therefore kerp. is abelian. 11m. §Aut(KIOt(K)) and since K is of class 2 it is regular. Hence corollary 11.1.5(a) 19 is applicable and lEImp] E logzs/t. Therefore lEAut(G|0t)] E logZZS/t. Furthermore , Aut (GIOt) /Aut (G > K. > 1) 6‘ Aut (Klflt (K)) . 8-1: But eprut(KJOt(K)) E by theorem 11.1.7(a). By lemma 1.2.2 P s ZS-t eprut(G > K.<flt> > 1) E p . Therefore eprut (GIOt) E p (b) We have again that Aut (Gl) /Aut (G > <flt> -K > 1) éAut(K|Qt (10). By theorem 11.1.7(b) , 3(s-t) 4s-3t eprut(KJOt(K)) I 2 Therefore eprut(G1) E 2 Corollaryg11.2.2 (a) If G is a p-group (p odd) and if III\ at S Z(G), then 1(G) log24s/t (b) If G is a 2-group and if at E Z(G), t > 1, then K(G) E log24(s-l)/(t-l). Proof (a) and (b) Since at EI'Z(G), G/Z CAut(G|Ot). We note that if A is maximal subject to being normal, abelian of exponent pt in G, then by Alperin [2] QtECG(A)] = A E Z(CG(A)). We may then, using the corollary above, set a bound on the derived length of CG(A). In fact 1(CG(A)) E log24s/t in the case p is odd and 1(CG(A)) E 10g 4(s-l/t-1) if p = 2, t > 1. Corollary 11.2.3 (a) If G is a p-group (p odd), expG = p5 and if A is maximal subject to being normal, abelian of exponent pt, then AEAut(G|A)] E log24s/t and eprut(G‘A) E p3s-t (b) If G is a 2-group, expG = ZS and A is as above, then l[Aut(G|A)] E log24(s-l/t-l) t > 1 and eprut(G|A) .s. 25‘3'3t t > 1. ‘Prggf (a) Let u: Aut(G‘A) ~.Aut(CG(A)) where u is the restriction. keru is abelian since CG(CG(A)) E CG(A) Inu E Aut(CG(A)|A). But OtCCG(A)] = A by Alperin [2]. Thus, by the previous theorem lEqu] E logZZS/t. Hence X[Aut(G‘A)] E log24s/t. 20 Furthermore Aut(GIA)/Aut(G > CG(A) > 1) c Aut(CG(A)lA). Therefore eprut(G|A) E p38-t (b) is proved similarly using (b) of the previous theorem. When A is maximal subject to being normal, abelian of exponent pt, we may obtain a finer bound on Aut(GiA) if we know that the exponent of G' is no greater than the exponent of A. We then have the following Theorem 11.2.4 (a) Let G be a p-group of odd order. Let expG = p8 and expG' f pn where pn'< ps. Let A be maximal sub- ject to being normal abelian of exponent pt. Then >.[Aut(G|A)] a logZZS/t if r a n. (b) Let G be a 2-group and A as in (a). Let expG' = 2n where 2n1< 28. Then erut(GIA)] E log22(s-l/t-l) if t E max(n,2). .Prggf (a) Let u: Aut(GIA) fl Aut(CG(A)|A) where u is the restriction mapping. By a result of Alperin.[2] 0t(CG(A)) = A. On the other hand, exp[CG(A)]' E pn E pt. Hence [CG(A)]' E A. But clearly A E Z(CGCA)) and therefore CGCA) is of class 2. Thus by corollary 11.1.5(a), the derived length of Aut(CG(A)IA) is no greater than logzs/t. Hence, since keru is abelian, 1[Aut(G|A)] a log228/t. (b) The proof is similar and we appeal to corollary 11.1.5(b) to obtain lEAut(CG(A)IA)] E log2(s-l/t-l). Thus 1[Aur(c|A)] e log22(s-1/t-l). We pause to give two simple examples which will illustrate the foregoing theory. 21 Example 1 Consider the group generated by P and Q_ with defining relations P9 = 1, Q9 = l, Q-lPQ = P4. This is a group of order 81 of class 2. Hence it is regular. A simple calculation shows that Z(G) = ®. It is easy to show that O = Z(G). 1 We may then apply theorem II.l.4(a) to obtain clAut(G101) 5 2-1 = 1. Furthermore c6611) = G and we know that clECG(01)] 5 2 by one of our results. Since ch = 2, we see that our bound is actually attained in this case. Example 2 Consider the group G generated by P,Q and R with the following relations: P9= 1, Q3 = 1, R3 = 1, [P,Q] = 1, [P,R] = Q and [R,Q] = P3. This is a group of order 81 and each element of group is expressible uniquely in the form PIQij. It is easily shown that [Pn,R] = [P,R]n for all n and l 3 3 "’ 3 . hence P ,R] = [P,R] = 1. Therefore

E Z(G) and in fact = 2(6). The class of the group is 3 and an easy calculation shows 1, 9(m-4) 3 (m-L)R3 that (PLQmR)3 = Q = l for all L and m. Thus PR.R = P, IPRI = 3 and RH = 3, but order of P is 9. Hence the group is not regular and we must use the results of section 2. Consider now the subgroup . This is clearly abelian and i j k E 3 normal of exponent 3. Suppose P Q R, CG(

). Then Q-IPIQJRkQ = Plech Hence Q-IRFQ = Rk. But from the relations, Q-IRQ = RP3. Therefore Q-leQ = Rszk = Rk. Therefore P3k =1, and k 3 O mod3. Thus the only elements in CGC

) are of the form Pin and thus (P3,Q> is maximal subject to being normal, abelian of exponent 3. 22 It may be shown that the mappings P T Pi, i = 1,4,7; Q v Q; R * PLQmR for arbitrary l and m are automorphisms which fix elementwise and furthermore these are the only such auto- morphisms. Thus IAut(Gl)| = 81. This is a non-abelian group of derived length 2. Applying corollary 11.2.3(a), since expG = 32, we get XEAut(GI)] é log28 = 3. Furthermore, CG() = ’ which is abelian. Again the bound obtained from our results for the derived length is 3. Section 3 In this section we obtain bounds for the de- rived length and exponent of Aut(GlA) where A is now maximal subject to being abelian and of exponent pt. We thus drop the hypothesis that A is normal. The bounds we shall obtain will depend not only on the exponent of A and of G, but also on the class of G. We note that because of Alperin's result [2], if A is maximal subject to being normal, abelian and of exponent pt, it is maximal subject to being abelian and of exponent pt. In this section we are therefore extending the results of section 2 to cover a larger class of subgroups. We start by proving the following: Lemma 11.3.1 Let G be a group and let H be a subgroup of 5 . = . . = G such that CG(H) H. Define HO H and inductively Hi NG(Hi-l)' If Hk = G, then AEAut(GIH)] g k. Furthermore if G is a p-group kt and epo = pt, then expEAut(G|H)] E p .; Proof If H = c, i.e. H< G, then Aut(GlH) = Aut(G) H> 1) l t by lemma 1.2.2, Aut(G > H:> 1) is abelian and eprut(G|H)=§ p . Assume inductively the result is true if Hn is the whole group where n'< k and suppose G is a group with HR = G. Then by induction 23 AlAut(Hk_1|H)] é k-l and eprut(Hk_1|H) é p(k-1)t. Consider .u; Aut(GlH)'* Aut(H Then Iml i Aut(Hk_1|H) and k-l)' keni = Aut(GlHk_1). But Hk_1 < G and CG(H E H.5 k-l) Hk-l' There- fore by lemma 1.2.2 Aut (cl Hk_1) is abelian and eprut(cl Hk_1) 5 pt E because CG(Hk-l) H. I“ Therefore xEAut(GlH)] k and eprut(G|H) é pkt. Lemma 11.3.2 Let ch c and let H.§ Gw Let Hi be as in the previous lemma. Then Hi 5 21, the ith center of G for all i. Thus H = G. c Proof Clearly H1_E'Zl. Assume inductively Hi E 21. Then - E E E ' ' Since [Hi’zi+l] Zi Hi,z1+1 Hi+1 and by induction we are done. Since Z = G, H = G. c c We note that in the above lemma, if CG(H) E H, Z1 a H and we may start the induction with HO 5 21. We may therefore show that Hc-1 = G. Corollary 11.3.3 Let G be a p-group and let A be maximal subject to being abelian. Suppose ch = c. Then AEAut(G|A)] E c-l and eprut(GlA) g pal-1)t where epr = pt. Egggf Since A is maximal abelian, CGCA) = A and the preceding two lemmas are applicable. Theorem 11.3.4 (a) Let G be a p-group of odd order. Let expG = p8 and let class G = c. Let A be maximal subject to being abelian and of exponent pt. Then X[Aut(GlA)] é logzs/t + c and eprut (cl A) E owing"t (b) Let G be a 2-group, expG = p8 and class C = c. Let A be as in (a) with t > 1. Then 'XEAUt(G|A)] é 1082(3'1/t'1) + C and eprut(G|A) E 2<3+C)S'3t. 24 £123; (a) Let u: Aut(Gl A) -. Aut(CG(A)lA) be the restriction mapping. Then keni = Aut(GlCGCA)) and Imp é Aut(CG(A)|A). But Ot(CG(A)) = A (this time we do not have to appeal to Alperin's result as in the case A normal) and hence x[Aut(CG(A)IA)] 3 logZZS/t and eprut(CG(A)IA) 5 pzs't by theorem 11.2.1(a). By lemmas 11.3.1 and 11.3.2, AEAut(G|cG(A)] E c-l and eprut(G|CG(A)) a p(°‘1)- . Hence lfAut(G|A)] é log223/t + c-l and eprut(G|A) 5 p =Qt, and not only in regular p-groups.) We begin by considering Ii: Aut(GlA)‘* Autélt) instead of CG(A). Thus A is maximal subject to being abelian in. 0t. Hence (c-l)t s-t by corollary 11.3.3 and eprut(GV7t) 5 p c-2 t+s p( ) . eprutGltlA) *- p by theorem 11.1.7(a). Thus eprut(G|A) g 25 We note that theorem 11.3.4 proves a theorem by N. Blackburn [4] which states that Aut(GIA) is a p-group. Section 4 In the previous sections we have been con- cerned with those subgroups of the automorphism group which fix elementwise certain subgroups of the group G. In this section we shall extend these results to include those subgroups of the auto- morphism group of G which do not move around "too much" the elements of certain subgroups of G. More specifically, we shall examine auto- morphism groups which induce power automorphisms on certain subgroups of G. -Definition 11.4.1 Let G’E Aut(G). Then a is said to be a power automorphism of G if for every H 5 G, H: = H,€Y is said to induce a power automorphism on a subgroup K of G if <1 re- stricted to K is a power automorphism of K. To illustrate the subsequent theory we shall work out an example in detail. .Notation Let H E G. Then Aut(GInH) denotes the group of those automorphisms inducing power automorphisms on H. Example Let G be the group generated by P and Q with relations P9 = Q3 = l; Q-lPQ*= P4. This is a group of order 27. Since Q-1P3Q = (Q-IPQ)3 = P12 = P3, . Let a be any automorphism of G inducing a power automorphism on '

. Then a: P 4 P1, 1 = l,2,4,5,7,8 and suppose a: Q “ PLQm, where m is either 1 or 2. Since a is an In - o 1 4- automorphism Q tnPiQm = P41 . Therefore P4 = P 1 and hence 4T 5 4i mod9. Thus m = 1. Furthermore IPLQI = 3 and 26 *PL = PLQQ-1. Therefore IPLI = 3 and.L E O mod3. .Thus P * Pi, i = l,2,4,5,7,8 Q * PLQ, L = 0,3,6 give rise to all the auto- morphisms inducing power automorphisms on ‘

; Thus lAut(Gifi

)l = 18. It is easily shown that .Aut(Glfi

) is non- abelian. ,Furthermore, IAut(Gi

)| = 3 and [Aut(Glfi

)]' = Aut(G|

). Thus Aut(G|fi

)/Aut(GL

) 2 C6’ the cyclic group of order 6. We shall show in the theorems below that this example is typical of the general case. We first prove some lemmas of a general nature. Lemma 11.4.1 Let HE G. Then [Aut(GInH)]' a Aut(GIH). 2322; Let 0,8 6 Aut(G H). Then if ix 6 H, 1et r“ = x , -1 . -1 . B j a i B = xi x »= x . Clearly x = x and x where x[a,B] g xi'j’ij = 11' e jj' e lmodlx'. Thus Corollary,11.4.2 (a) Let G be a p-group of odd order and let expG = p8. Then llAut(Gant)] é log24s/t. (b) Let G be a 2-group, expG = 28. Then 1EAut(Gant)] é log24(s~l/t-l) where t > 1. nggf_ Lemma 11.4.1 and theorem 11.2.1(a) and (b). Lemma 11.4.3 (a) Let G be a p-group of odd order and let t-l _ H-E G and epo = pt. Then if’ a E Aut(Glflh), the semi- direct product of G with 1<1>. Let x E H. Then NB () /CB (x) ‘3 Aut () 5’ C But 01 6 NB () . Hence Pt“1 pt’1(p-l) a E CB(x). This is true for all x E H and therefore t-l up (p'l) e Aut(GIH). 27 (b) is proved similarly except that if lxl = 2r, Aut() z Czr-l if r = l or r = 2 and Aut6) z 02‘? C21.“2 if r~E 3. .Corollary 11.4.4 (a) Let G be a p-group of odd order, expG = p8. Then eprut(Gl€pt) é p(28-1).(p-1). (b) Let G be a 2-group, expG = 28. Then eprut(G|fi) 5 248-2t-1. 2322: Lemma 11.4.1 and theorem 11.2.1(a) and (b). Lemma 11.4.5 Let G be a p-group, H a non-abelian subgroup of G. Then Aut(GLnH)/Aut(GIH) is an abelian p-group. 'Prggf Let u: Aut(GlnH) T Aut(H) be the restriction map. Then by a result of Cooper [6], Imu is an abelian p-group. Corollary 11.4.6 Let G be a p-group of odd order. Then if 'Ot is non-abelian, Aut(GifiQt) is a p-group of exponent at most p23”1 where expG = p3. Furthermore, whether p is odd or even, if {71 is non-abelian Aut(GLéal) = Aut(Glol). 2523; The first statement is proved using corollary 11.4.4(a) and lemma 11.4.5. To prove the second statement we use a result of Cooper's, namely, that a power automorphism is central [6]. Thus, suppose Aut(G'nfll) e Aut(clfll) and let a e Aut(GInfll) - Aut(G‘fll). Then there must be an element of order p, say x, such that x-lxa # 1. But x-lmy 6 2621) by Cooper's result quoted above and x-lx? E . Hence x 6 2(01). Thus any element of order p which is moved lies in the center of 01. Suppose if possible that y is an element of order p and y 9 201). Then xy E Z(Ol). .Thus a a (xy)a = xy = x y and hence x = x, a contradiction. Therefore .01 = 2671). By contraposition, the statement in the corollary follows. 28 We now quote a theorem due to Gascthz [9]. Theorem 11.4.7 If G is a finite abelian group and a is a power automorphism of G, then there exists a fixed positive integer m such that g1 = gm for all g E G. Lemma 11.4.8 (a) Let G be a p-group of odd order and H an abelian subgroup of G, epo = pt. Then Aut (GI TrH) /Aut (GI H) <9 Cpt‘1(p-1) (b) Let G be a 2-group and H as in (a). Then Aut(G H)/Aut(G|H)8c a c if t e 3 and n 2 2t-2 Aut(GlnH)/Aut(G| H) asc if t = 1 or 2. 2t-l Proof (a) Let a E Aut(GlnH). Then by theorem 11.4.7, there exists a positive integer m such that ha = hm for all h E H. Consider therefore the mapping 1*: Aut(GinH) * J tg*wheré J t-is the ring P t of integers modp , given byllta 4 m modpt. This is clearly a homo- morphism into the multiplicative group of units of J t' But the P multiplicative group of units of J t is cyclic of order pt-1(p-l). P Furthermore, keru = Aut (GIH) and hence Aut (GInH) /AUt (G'H) QC t-l P (p-l) (b) is proved similarly except that the multiplicative group of units of J t is cyclic of order Zt-l if t = 1 or 2 and 2 . ' 2 is of the form C2 3 Czt_2 if t 3. Example This will illustrate that in general we cannot con- clude that Aut(GLnH)/Aut(GlH) is a cyclic group if G is a 2-group. Let G = ) first. If P * Pi, Q“ PLQ is an automorphism, then i = l,3,5,7. 2 L 2 Furthermore, it is easily shown that [P Q] = Q and that the ' L remaining relations are satisfied. Thus P E P1, Q a P Q, 29 i = 1,3,5,7, L = 0,1,2,3,4,5,6,7 constitute Aut(Glfi). Hence IAut(Glfi

)l = 32. Also P d P, Q a PLQ, L = 0,...,7 give rise to Aut(Gl

) and so lAut(Gl

)l = 8. Therefore Aut(GIfi

)/Aut(GL

) is of order 4. But if aEP a Pi, Q a PLQ] is an automorphism of Aut(G|fi

), 02 E Aut(GL) since 12 E 1 mod 8 for i = l,3,5,7. Hence Aut(Gln

)/Aut(Gl

) 2 C2 ® C2. Corollary 11.4.9 (a) Let G be a p-group of odd order, expG = p8 and let ch = Let A be maximal subject to being c abelian and of exponent pt. Then P, the p-sylow subgroup of Aut (GlnA) is normal in Aut(d 1TA), Aut (GlnA) = P.C where C is cyclic, |C||(p-l) and KEAut(GinA)] é logZZS/t + c. (b) Let G be a 2-group, expG = p8 and let ch = e. Let A be as above with t > 1. Then Aut(GlnA) is a 2-group and llAut(G|fiA)] a log22(s-1/t-l) + c. Proof (a) Aut(GlfiA)/Aut(G]A) etc: by lemma II.4.8(a). Pt'1(p-1) But Aut(GIA) is a p-group containing [Aut(GlfiA)]'. Hence p-sylow 'subgroup P of Aut(GlfiA) is normal and therefore it possesses a p-complement C. Therefore Aut(G|fiA) = P.C. Furthermore, lEAut(G‘fiA)] a 1og22s/t + c by theorem 11.3.4. (b) is proved similarly using (b) of lemma II.4.8(b) and theorem 11.3.4. Section 5 In this section we investigate Aut(GlH) where G is a p-group and H is a maximal subgroup. We shall find application of these results in the next chapter. Lemma 11.5.1 Let G be a p-group (p-odd or even), H a maximal subgroup. Then llAut(G|H)] E 2 and Aut(G'H) = Aut(G > H > l)-C where C is cyclic, ‘C||(p-1). 3O * Proof Let u: Aut(GIH) *.Aut(G/H) be the mapping u: 0 fl 0 * to where (ngy= g H. Clearly u is a homomorphism and keru = Aut(G > H > 1) while 11111 a Aut(G/H) : Thus cp_1. Aut(GIH)[Aut(G > H > l)<5‘CP_1. The lemma then follows easily. We note that if G is a 2-group, Aut(GIH) = Aut(G > H > 1) and therefore Aut(GIH) is an abelian 2-group. Corollary 11.5.2 Let G be a 2-group and let G = G >...> G 0 m O,1,...,(m-l). be a chain of subgroups such that lGizG = 2 for i i+1l — a ' = 2. Let Ak {a E Aut(G)|Gi Gi’ 1 1,...,k} and BL Aut(GIGL). . ' - = S Then, Ak'O Bk is a 2 group, k 1,...,m and lEAk O Bk] - k. Proof A1 n 131 = Aut(GIGI) = Aut(G > G > 1) by the lemma and 1 hence for k-= l, the corollary is true. Assume inductively that the result is true for k. Consider u: Ak+1O B k+1 ~.Aut(Gk) where u is the restriction map. Then keru ='Ak+ IO Bk+1O Aut(GIGk) = a Ak+l n Bk+ Ak n Bk and Inn E Aut(Gk/G ). By induction Ak n 13k is a 2-group and k+l AEAk O Bk] 5 k. Furthermore, 1m; is an abelian 2-group by the previous lemma. Therefore Ak+ 1O B/Ak O Bk is an abelian 2-group and the result is proved for k + l and by induction we are done. Theorem 11.5.3 Let H be a maximal subgroup of a regular p-group G (p odd). Then either Aut(GIH) is an abelian p-group or G = H ® in which case if oz 6 Aut (G‘H) either ap-l = l or a is of p-power order. Proof There are two possibilities: either (a) Co (H) a l or (b) CO (H) E H. l 31 Suppose (a) holds. Then. 01 O CG(H) E H. Let g E 01 and a E Aut(GIH). Then by Lemma 1.2.2, g-lga E CG(H) O 01 E 01 O H. Consider therefore the mapping u: Aut (GIH) -0 Aut (01) where u. is the restriction. 11111 i Aut(flllfl1 O H) = Aut(fi1 > 01 O H > 1) since g-lga E 01 O H for all g E 01. Furthermore, keru =.Aut(6101) which is a p-group by 11.3.4. Hence Aut(GlH) is a p-group and by lemma 11.5.1 we have Aut(GlH) = Aut(G > H > 1) which is abelian. Suppose (b) holds. Then there is an element x E Ch1(H) - H. Therefore G -= H ® . Now let a E Aut(GIH). Then xa = xiha, say, with ha 6 H. We then get ap'l _ ip'1 1+i+i2+...+ip'2 x - x h a . .p-2 ++ooo+ -1 = x“: 1 1 since ip .5 1 mod p. Now if i = l, a E Aut(G > H > 1) and a is of p-power order. On gl-ip'l) ' ap'l (l-i) the other hand, if I. # 1, X = X h = x since |ha| = P and p divides (l-ip-1)/(l-i). Hence in case (b) each element of Aut(GIH) is either a power of a prime or has order dividing (p-l). CHAPTER III In this chapter we investigate Aut(G) where G is a p-group of maximal class. This class of groups was investigated by N. Black- burn and we quote a number of results which will be found in his paper [5]. Definition 111.1.1 Let G be a p-group. Then G is of maximal class if the nilpotence class of G is the greatest compatible with its order. It is proved in [5] that if IGI = pm and c is of maximal class, then cJG = m-l. Furthermore, it can be shown that G/T2 = p2 and G/r2 is elementary abelian. This implies that Ifizfi+ll = p for i = 2,...,m-l. Definition 111.1.2 Let G be a p-group of maximal class, |G| = pm, m > 3. Define F1 by I'lll‘4 = CG 4(IE/11). Clearly 'F1 is a characteristic subgroup and it can be shown that IG:F1| = p. We thus have a chain of characteristic subgroups G = To > F1 >...> F > Fm = 1, each Pi maximal in F. Further- 1-1. 2 more, it is easy to show that any normal subgroup of G of index p m-l or greater is one of the Ti. We first prove a few general lemmas which will be used in the sequel. Lemma 111.1.1 If G is generated by x1,... n - a a E AutG such that xilxi E H, Hid G for i = 1,...,n, then 32 33 x'lr“ e H for all x e c. k Proof Let x = H yi where yi E [x1,...,xn]. Then if k = 1, i=1 x 1r“ E H by hypothesis. Assume the result is true if kt< r and r let x = H y,. Then ._ 1 1-1 -1 a -l a x x = (y1---yr_1yr) (ylo--yr) _ -1 -1 a a _ yr (yl...yr_1) (yl...yr_1) yr . . . -1 a By induction (yl...Yr_1) (y1"°yr-l) = h E H. Therefore a -1.a a x x = yr yrhlh,yr] E H since H*< G. Hence by induction, we are done. Lemma 111.1.2 Let G be a nilpotent group. Then if 0' a E Aut(G.F2), [x1,...,xm] - [x1,...,xm] modFm+1. Proof When m = l, the result is true by hypothesis. Assume inductively that for simple commutators of weight less than m, the a _ a a result is true. Now [x1,...,xm_1,xmg - [[x1,...,xm_1] ,xm]. But a ... = ... - d d [x1, ,xm_1] [x1, ,xm_1]cm, where cm E Fm by in uction an a = 1" . xm xmc2 where c2 E 2 by hypothesis Thus 0’ [x1,...,xmg - [[x1,...,xm_1]cm,xmc2] = [[x1,...,xm_1]cm,c2][[x1,...,xm_1]cm,xm][[x1,...,xm_1]cm,xm,c2] by lemma 1.1.2. We now use the fact that [Fi’rj] S ri+j for all i,j = 2,3,... (Theorem 1.1.1). Then T . [[X1,...,Xm_1]§m,62] E rm+1 and [[x1’°°"xm-1]Cm’xm’cz] E m+2 a ... = ... F Therefore [xl’ ,xmg [[Xls ’xm-IJCm,xm]Cm+1’Cm+1 E m+1 34 = [x1,...,xm][x1,...,xm,cm][cm,xx]crn+1 = [x -,X ]c' ' E F m 1". m+l’cm+l m+l° Hence by induction the result follows. We recall that Pk is generated by simple commutators of weight k. Using this fact and lemma 111.1.1 we obtain the following: Theorem 111.1.3 Let G be a finite p-group. Then Aut (6:12) = Aut (G > 1"2 >...> 1). We prove a final lemma before considering Aut(G), G of maximal class. Lemma 111.1.4 Let E be an elementary abelian group of order p2 generated by P and Q. The group A of automorphisms given by P—oP i 1,...,(p‘1) 0.-~.(p-1) a L Q QP L is of order p(p-l) and of derived length 2. Proof The group A of automorphisms is clearly of order p(p-l). Furthermore, it is isomorphic to the group of non-singular matrices of the form (2 (i) with entries from a Galois field of . -l O i O p elements. Now it is easily checked that (ll 1) = (4% 1) where ii' 5 1 mod p. Then (1:46 3 sz 3))(2 $(i (l) = G; (1)) Now c (i) has order p and therefore lies in the unique sylow p-subgroup which is cyclic of order p. Hence A' is abelian and A is metabelian. We are now in a position to prove that if G is a p-group of maximal class, |G| = pm, m > 3, then Aut(G) is solvable and we may 35 obtain a bound on the derived length together with some information about the order of Aut(G). We note that if m = 3, Aut(G) is not necessarily solvable as can be seen from the following example: P = 1; [ml = [P.R] = 1; Let c = Aut(G) is not solvable as can be seen as follows: If Aut(G) were solvable, so would Aut(GIZ). Consider the mapping P-+P a B Q_.Q1R1 0' B R-°Q2R2 Under what circumstances can this be extended to an automorphism of a‘ BI a B G? It can be extended if and only if [Q 1 R , Q 2 R 2] = P. Since ch = 2, the above hold if and only if a B 4! B [Q,R] l 2 2 1 = P i.e. if and only if 0182 - 0281 5 1 mod p. Consider now the homo- morphism u: Aut(GIZ) dlAut(G/Z). Now G/Z is elementary abelian of order p2 and hence Aut(G/Z) z GL2(k) where k is a Galois field of p elements. Further, by what we showed above, Inn z SL2(k) and SL2(k)/Z(Slq(k)) is the projective unimodular group which is simple if p > 3 [3]. Hence Aut(G) cannot be solvable. Before proving the next theorem, we calculate IAut(G)| where |G| = p3 and G is non-abelian. When p = 2, it is well-known that IAut(G)| = 24 or 8. We therefore perform the calculations for p odd. 3 There are two non-abelian groups of order p , viz. 36 (i) The one of exponent p generated by ~P, Q and R with the following generating relations: Pp = Q" = R" = 1.R'1QR = QP.R'1PR = P. Q'IPQ = P and (ii) The one of exponent p generated by P and Q with 2 - 1 generating relations Pp = Qp = 1, Q 1PQ = p +p. We calculate (i) first. Z(G) =

and thus any automorphism must be of the form i . P-* P , i = 1,...,)p-l) Q -. PLinls,L = 0,...,p - 1 Its! R a PL Qm R , {.= 0,...,p - 1. We also require l I I . [PLQr'hs, P Qtn Rs ] = P1. Since G is of class 2, this condition is equivalent to [Q.R]'“S'[R.Q]S“" = pi ms'-s ' i Therefore P m = P and hence ms - sm 5 i mod p. Now if neither m nor 5 is zero, we have p choices for the pair (m',s'). If m = O and 3% 0 or if m f 0 and s = 0 we get p choices for (m',s') in each case and thus we have 3 3 2 3 2 3 2 3 (9-1) p + 2(p-1) p = (13-1) p (p-1+2) = (13-1) p (9+1) choices and each of these gives rise to an automorphism provided ! I E . Now calculating modulo Z(G) (QmRS))‘(Qm'RS ')u. = le-mm'Rks-lus' and to obtain Q(modulo Z(G)) we must solve the equations 37 Km +-um' 1 KS +-us' = 0 in GF(p). m m' s s') f 0 and the equations have a solution. Thus But det ( I I Q = (qus)l(Qm Rs )“ mod Z(G) for some )1 and 11.. Similarly for 3 R. Hence Aut(G) = (p-l)2p (p+l). Now we calculate (ii). Let P-0 Pin i! jl Q‘* P‘ Q give rise to an automorphism (PIQJ)p = PiRQJPEQj,Pi](g) = P1p # 1. Hence 1 $.0 mod p. Similarly i' E 0 mod p. The following relation must also hold. [PinJi'Qj'] = (Piqbp = Pip Therefore [P,Q]ij"j"j = Pip I_I - Hence Pp 3. Then (i) P = Aut(G > F1 >...> Fm- > 1) is the p-sylow subgroup 1 of Aut (G). Hence P <1 Aut (G) and clP E m-l. (ii) Aut(G) is solvable and lEAut(c)] =1 log28(m-l) (iii) Aut(G) = PL where |L||(p-1)2 and Pm é IPI a p2m-3 38 (b) Let G be a 2-group of maximal class, IG| = 2m, m > 3. Then Aut (G) = Aut(G > F1 >...> Fm- > 1) and hence l Aut (G) is a 2-group of class not greater than m-l. Furthermore 2‘n .s. |Aut(c)| a 22"”. Proof (a) (i) Let P be a p-sylow subgroup of Aut(G). Clearly P induces the identity automorphism on Fm-l' Assume inductively that P stabilizes I‘m_k.>...>1"m_1 > 1. Consider : -» F as _ B P Aut(Fm_k_1/ m-k) Cp-l where B is the obvious homo morphism. Then ImB = l and P stabilizes > ...> > . ' ' rm-k-l Fm-k > I‘m_1 1 Hence by induction, P stabilizes c>l" >...>F >1. Therefore P=Aut(G>F >...>F l m-l l m- and by P. Hall [13], clP a m-l. > 1 1) (ii) and (iii). Let a: Aut(G) -» Aut(G/1‘1) be the mapping * * defined by B: a -’ or where (gF1)a = gafl'B is a homomorphism and Imfi G C kerB = Aut (G;F1) . p-l’ ConSider now 0: Aut (G;I"1) -* Aut (G/Tz). G/I‘2 is elementary abelian of order p2. Thus G/I‘2 = Fl/FZ 8 14/?2 for some M E G. Since 1'“1 = F1 for all a E Aut(G), Imc is a subgroup of Aut (G/Fz): each element of which induces a power automorphism on Fl/Fz. Thus if G/I"2 = ®, a 6 F1, the automorphisms of 11110 are of I 2 I 2 3 ' ° ° 3p bI‘2 -~ ai’bml‘2 L '= 0,...,p-1; \m = 1,...,p-l. - -1 But we further have that acbmb 1 E F1. Therefore bm E F1 which implies m = 1. Hence the automorphisms of ImO are of the form 39 i . aFZ ~ a F2 1 = 1,...,p-1 t bf'z abr‘2 t 0,...,p-l Therefore lemma 111.1.4 is applicable and we have |1m0l|p(p-l) and lEImU] E 2. Furthermore, kero = Aut(G;F2) which by theorem 111.1.5 is a p-group of class at most m-2 (actually, the class is exactly m-2 since G/Z GAut(G;F2)). Now IAut(G)I =IAut(G;I‘1)I-t where tI(p-1) and IAut(G;F1)| = |Aut(G;F2)|-kt* where k = l or p and t*|(p-l). Therefore IAut(G}| = pns where sl(p-l)2. To prove that m E n I 2m - 3 we observe first that since G/Z‘i’P, IPI E pm as every p-group has an outer automorphism of p- power order [10]. To obtain the upper bound we argue by induction. When m = 3, from the discussion preceding the theorem, IPI E 2.3 - 3. Thus assume inductively that ‘P‘ E p2m-3 if lGl B: P ~.Aut(G/Z) where P is the p-sylow subgroup of Aut(G). By induction |B(P)I E p2m_3 and kerB = Aut(G > Z > 1). Now since Pm, m > 3. Let IG' = pm+1. Then |G/Z| = pm. Consider G is of maximal class, it is clearly purely non-abelian. By a 2 result of J. Adney and T. Yen [l], IAut(G;Z)I = IHom(G/F2,Z)| = p . Therefore IPI E p2(m+l)-3 and the result follows by induction. Finally, consider Aut(G) EAut(G;I"1) EAut(G;1"2). Aut(G)/Aut(G;F is abelian and lEAut(G;F1)/Aut(G;F2)] E 2. 1) Furthermore, 1[Aut(G;F2)] E log2(m-l) and therefore IEAut(G)] E log28(m-l). Now lot L be a p-complement of Aut(G). We have Aut(G) = P.L. (b) To prove Aut(G) is a 2-group, we proceed in exactly the same way as above. 40 Corollary 111.1.6 If G is a 3-group of maximal class, C = 3m, m.> 3, Aut(G) is a split extension of 3-group by an abelian 2-group of order 2 or 4, or a 3-group. NCHATION Ot(G) set of elements of G of order not greater t than p <flt(G)> group generated by Ot(G) [3t(G) set of elements which are pt-th powers of elements in G Aut(GlH) group of automorphisms of G fixing element- wise Ii Aut(G3H) group of H-automorphisms of G (cf. Definition I.1.3(a)) Aut(G > G1 >...> Gm) group of automorphisms stabilizing G > C1 >...> Gm (cf. Definition I.1.3(b)) c1(G) nilpotent class of G X(G) derived length of G exp(G) lowest common multiple of the orders of elements of G k . k C(p ),C k cyclic group of order p P CG(H) centralizer of H in G NG(H) normalizer of H in G Aut(GI H) the group of automorphisms inducing a power fl automorphism on H * * (;¢9(} G contains a subgroup isomorphic to G 41 BIBLIOGRAPHY 10. 11. 12. 13. 14. BIBLIOGRAPHY Adney, J. and T. Yen, Automorphisms of a p-group, Illinois Journal of Math, Vol. 9, No. 1, March 1965. Alperin, J., Centralizers of abelian normal subgroups of p- groups, Journal of Algebra 1, 1964. Artin, E., Geometric Algebra, Interscience Tracts in Pure and Applied Methematics (1957). Blackburn, N., Automorphisms of finite p-groups, Journal of Algebra 3 (1966). Blackburn, N., On a special class of p-groups, Acta Math. 100 (1958). Cooper, C.D.H., Power automorphisms of a group, To appear in (Math. Zeitschr. Feit, W. and J. Thompson, Solvability of groups of odd order, Pac. J. Math. 13 (1963). Fuchs, L., Abelian Groups, Pergamon Press, 1960. Gaschgtz, W., Gruppen in denen das Normalteilersein transitiv ist, J. Reine und Angewandte Math 198 (1957). Gaschgtz, W., Nichtabelsche p-Gruppen besitzen aussere p- Automorphismen, J. of Algebra 4 (1966). Hall, M., The Theory of Groups, Macmillan, 1959. Hall, P., A contribution to the theory of groups of prime power order, Proc. London Math. Soc. 36 (1933). Hall, P., Some sufficient conditions for a group to be nil- potent, Illinois Journal of Math. 2 (1958). Scott, W.R., Group Theory, Prentice Hall, 1964. 42