LIBRARY Michigan Stab University This is to certify that the thesis entitled Normal p-Subgroups in the Automorphism Group of a Finite p-Group presented by Dawn Rickard Shapiro has been accepted towards fulfillment of the requirements for PhoDo degree in Mathematics aJor professor Date 9/.3/7? 0-7639 NORMAL p-SUBGROUPS IN THE AUTOMORPHISM GROUP OF A FINITE p-GROUP BY Dawn Rickard Shapiro A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1978 ABSTRACT NORMAL p-SUBGROUPS IN THE AUTOMORPHISM GROUP OF A FINITE p-GROUP BY Dawn Rickard Shapiro Let G be a finite p-group, p an odd prime. The primary purpose of this dissertation is to study the non- trivial normal p—subgroups of the group Out G = Aut G/Inn G of outer automorphisms of the group G. Given a chain of subgroups s:G = G '2 G1 2,..2 G = 1, define Stab(s) by Stab(s) = {a 6 Aut G[(giGi+1)a = 916 for all i+1' gi eGi' 1:091'2’00091‘1-130 In a finite p—group G, the stability group of a characteristic chain is a normal p—subgroup of Aut G. Also, if A is a normal p-subgroup of Aut G, then A determines a unique characteristic chain in G which it stabilizes. When B‘g Stab(s), 5 denotes the closure of B, where 13 = Stab(G 2 [6,8] 2 [G,B,B] 2...2 1), and B is said to be closed if B = 5. Moreover, if B 3 Aut G. then fiig Aut G. We use these facts in our efforts to determine normal p-subgroups of Aut G. Dawn Rickard Shapiro In Chapter I, we prove the following embedding . theorems: (1) Let G ‘be a finite group, G = HK where H g_Z(G) is cyclic of order pn, p a prime, and K is normal in G. Suppose H n K is characteristic in K. Then Aut K embeds isomorphically in Aut G. (2) Let G be a finite group. Let H and K be characteristic subgroups of G such that H'2‘K. There exists a homomorphism a a &' mapping Aut G to Aut H/K such that a 6 CAut(3(H/K) if and only if a = 1Aut H/K' We also show that an extra—special p-group of exponent p, p an odd prime, contains no proper characteristic extra— special subgroups. Chapter II deals with p-groups of Hall type, where p is an odd prime. First we derive some basic prOperties of G, then we show that the maximal normal p-subgroup of Aut G, Op(Aut GL is the group of central automorphisms of G. Finally, we determine necessary and sufficient conditions for a non—trivial normal p-subgroup of Aut G to be closed as a stability group and, when G is not extra-special, show that a non-trivial normal p-subgroup of Aut G which is closed as a stability group, proPerly contains the group of inner automorphisms of G. Dawn Rickard Shapiro In the final chapter, Chapter III, we show that under certain conditions, lifting a stability group of a chain in a normal subgroup, will produce a normal and even closed stability group in the automorphism group of the whole group. Included in this chapter are results which can be used to Obtain closed abelian stability groups. To My Parents, Theodore and Dorcas Rickard and My Husband, Mike Shapiro ii ACKNOWLEDGMENT I would like to acknowledge the advice and support Professors Robert D. Carnes, S. Patrick Cassens and Marshall D. Hestenes have given me throughout my studies. A special word of thanks is due my thesis advisor Professor J.E. Adney for his encouragement and guidance not only in the preparation of this thesis but also throughout my graduate career. I‘am especially appreciative of the many hours he gave so freely from his already overloaded schedule. I would also like to thank Professors Edward Ingraham and Richard Phillips for their helpful comments and suggestions. Finally, I am very grateful to Mary Reynolds for her careful preparation of the final text. iii INTRODUCTION CHAPTER I: CHAPTER II: CHAPTER III: BIBLIOGRAPHY TABLE OF CONTENTS Preliminary Definitions and Results p-Groups of Hall Type Closed Subgroups and Lifting of Automorphism Groups iv 17 34 46 I. II. Relations: IA.flw\ V\ [A Qperations: C > X0. [XIY] [x,a] [H-A] n A [HIA’ o 0 05A] [H :K] H/K HK INDEX OF NOTATION Is a subgroup of Is a prOper subgroup of Is a normal subgroup of Is a normal p-subgroup of Is isomorphic to Is an element of Is congruent to The subgroup generated by The image of X under the mapping a -1 -l X Y XY -1 x xa Subgroup generated by all [h,d], h e H, a €.A. n—l r—’\—~ [[H,A,...,A],A] The index of K in H The cosets of K in H {hklh e H and k e K} III. H X K H 3 K H \.K Groups and Sets: Aut G Out G Inn G CH (K) NH (K) CAut G (H/K) G ’ 9(G) Z(G) Hom(H,K) Ker m The direct product of H and K The split extension of H by K The elements of H not in K The order of the element h The number of elements in H The exponent of H The class of H The mapping h 4 h for all h E H The mapping a restricted to H The greatest common divisor of a and b a divides b Product, not necessarily direct product The automorphism group of G The outer automorphism group of G The inner automorphism group of G The centralizer of K in H The normalizer of K in H [a e Aut GIh—lha e K for all h e H} The commutator subgroup of G The Frattini of G The center of G The set of homomorphisms of H into K The kernel of the homomorphism m vi INTRODUCTION In 1966, Gaschutz [3] showed that the outer automorphism group of any finite non-abelian p—group, p a prime, possesses a non—trivial p-subgroup. Later, in 1976, Schmid [7] determined all finite p-groups for which the group of outer automorphisms contains a non-trivial normal p-subgroup. This dissertation is a direct result of efforts to characterize the normal p-subgroups of Aut G, and thus of Out G, when G is a non-abelian Stability groups and subgroups of stability groups play an important role in this search for normal p-subgroups. The examples of normal p-subgroups in the outer automorphism group of a finite p-group which were Obtained by Schmid in [7] are stability groups. Moreover, in 1974, Bertelsen [2] showed that stability groups and their subgroups offer a viable means by which we may obtain normal p-subgroups in the automorphism group of a finite p-group. He shows that if a normal p—subgroup A of Aut G is a subgroup of a stability group then A, the closure of A, is a stability group which is a normal p-subgroup of Aut G. When A is a subgroup of a stability group and A = A then A is said to be closed. We ask how the non-trivial normal p-subgroups of Out G and the normal closed stability groups of Aut G are related. Chapter I contains definitions and results which are used in later chapters. We derive some embedding theorems which prove to be useful in the study of p—groups of Hall type, where p is an odd prime, and show that an extra- special p-group of exponent p, where p is odd, contains no proper characteristic extra-special subgroups. In Chapter II we consider p-groups G of Hall type, where p is an odd prime. After deriving some basic facts about G, we show that the maximal normal p—subgroup of Aut G is the group of central automorphisms. We Obtain a partial answer to our question about the relationship between non-trivial normal p—subgroups of Out G and the normal closed stability groups of Aut G. When G is of Hall type but not extra—special, we determine necessary and sufficient conditions for a non-trivial normal p—subgroup of Aut G to be closed as a stability group and show that a non-trivial normal p-subgroup of Aut G which is closed as a stability group preperly contains the group of inner automorphisms of G. The final chapter, Chapter III, consists primarily of two topics: (1) extending (lifting) a group of automorphisms of a normal subgroup to a group of automorphisms of the whole group and (2) closed stability groups. we consider groups G which are split extensions and show that under certain conditions, a stability group of a chain in a normal subgroup can be lifted to a normal and even closed stability group in the automorphism group of the whole group. We also include results which can be used to obtain closed abelian stability groups. CHAPTER I PRELIMINARY DEFINITIONS AND RESULTS This chapter contains definitions and results which are used in later chapters. All groups are assumed to be finite. Definition 1.1. Let H and K be subgroups of an arbitrary group G such that H 2_K. Define c:AutGm/K) by _ -l a cAutGm/K) — {a EAut Glh h EK for all h 6H}. Definition 1.2. Let s:G = GO-Z Gl-2"52 Gn = 1 be a chain of subgroups for an arbitrary group G. Define the stability group of s, denoted Stab(s), by Stab(s) = {a €.Aut G[(giGi+l)a = giG for all i+l gi eGi, 1=O,1,..o,n-1}. n-l With the above definitions, Stab(s) = F] C (G./G. ). i=0 AutG 1 1+1 Result 1.3 ([2],p.5). If each Gi is characteristic in G then Stab(s) gAut G. Definition 1.4. Let A g_Aut G. Set 'YGAO = G, and YGA1+1 = [YGA1,A], for i 2 0. Result 1.5. Using the notation of 1.2, let A g_Stab(s). Then (i) YGA1_<_Gi, i=0,l,...,n. (ii) The prime divisors of A are the same as the prime divisors of [G,A]. Proof: See Schmid [8]. The following remark is an immediate consequence of this result. Remark 1.6. If G is a finite p-group and s:G = GO-Z Gl-2°"2-Gn = 1, then Stab(s) is a p-group. Result 1.7 ([2],p.l6). Let G be a group. Let A 3 2(6) the center of G, c3 = 9A and a e Stab(G _>_ A _>_ 1). Then Stab(G 2 A Z 1) e: Hom(G/A,A) by the mapping a « fa: g 4 9-1 ga. Result 1.8 ([4],p.200). Let H be an abelian subgroup of an arbitrary group G. Let A = Stab(G 2.H‘2 1). Then A is abelian. Definition 1.9. Let A g_Aut G. We say A stabilizes a chain 5 if A < Stab(s). Following Schmid [8], 'we denote by TG the collection of subgroups of Aut G stabilizing a chain. We now define a concept introduced by Bertelsen in [2]. Definition 1.10. For .A E T , define E by G 0.2 YGRPZ--a2 VGAn(A), where n(A) is the first integer n(A) S:YGA such that YGA = 1. Define A, the closure of A, by A Stab(E). We say a stability group A is closed if A=A. Result 1.11 ([2],p.6). Let A 6 TG. Then (i) A g A. (ii) If A g_B < A, then yGBl = YGAl for all i. (iii) A = A. i (iv) If e 6 NA (A), then (ycAi)B = VGA . utCB (v) If A _<_1 Aut G, then A g Aut G. (vi) A and A have the same prime divisors. Remark 1.12. By 1.11 (iv), if A 6 TG and A g Aut G then YGA1 is characteristic in G for all i. Result 1.13 ([2], p.9). Let H g_§(G), the Frattini subgroup of G. If B = CAut<3(G/H) then B is a closed stability group. Definition 1.14. Let G be a p-group and A g_Aut G. A is said to be of K—type if (i) A is a p-group. (ii) A is normal in every p-Sylow of Aut G that contains A. (iii) A is the intersection of all the p-Sylows of Aut G that contain A. Result 1.15 ([2],p.21). If G is a p-group and A is of K-type then A is a closed stability group. Definition 1.16. Let G be any group. OP(G) will denote the unique maximal normal p-subgroup of G. Op(G) may be obtained by intersecting all p-Sylows of G. Hence Op(Aut G) is the smallest K-type stability group. Corollary 1.17. If G is a p—group and A = Op(Aut G) then A is a closed stability group. Definition 1.18. A finite non-abelian p-group G is extra-special if the center Z(G) is cyclic of order p and G/Z(G) is elementary abelian. Definition 1.19. If G is a pegroup, 1 . i Qi(G) = (x 6 Glxp = 1) and (51(G) = (xp Ix E G>. Result 1.20. Let G be a finite non-abelian p-group containing no non-cyclic characteristic abelian subgroups. Then the Frattini subgroup §(G) of G is cyclic. Moreover: (i) If p is odd, 01(G) is extra-special of exponent p and G = 01(G)Z(G). (ii) If p = 2, U = CG(§(G)) is of index at most 2 in G and one of the following holds: (a) G is extra-special. (b) G is a dihedral, semi—dihedral or generalized quaternion group. (c) §(G) = @(U) is of index 2 in the cyclic subgroup Z(U). This result is an unpublished theorem of P. Hall and can be verified by using the details given in the proof of Satz 111.13.10 in [5]. Definition 1.21. A group G satisfying the hypotheses of 1.20 will be called a p-group of Hall type. Result 1.22 (J. Thompson, [9]). Let G be a finite p-group. Given any maximal characteristic abelian subgroup Z of G, there exists a characteristic subgroup K of G such that CG(K) = Z(K) = Z and K/Z is an elementary abelian central factor of G. This is a modification of Lemma 3.7 in [9]. A proof may be found in [4], pages 185—186. Definition 1.23. A characteristic subgroup K of a p-grOup G is called a critical subgroup of G if CG(K) = Z(K) and K/Z(K) is an elementary abelian central factor of G. Corollary 1.24 (Schmid, [7]). A finite non-abelian p-group G is of Hall type if and only if the center of any critical subgroup of G is cyclic. We will later show that if G is of Hall type and p is odd then the only critical subgroup of G is G itself. Result 1.25 (Schmid, [7]). Suppose that G is a finite non—abelian p-group having a subgroup N such that Stab(G _>_ N _>_ 1) is equal to @(G) 3 NS Z(G) then c Inn G if and only if N = Z(G) is cyclic. In any case, C contains Inn G. Theorem 1.26. Let G = HK [where K.g G and H = is cyclic of order p“, p a prime. Suppose H's Z(G). If H rlK is characteristic in K then there exists a homomor- phism which embeds Aut K isomorphically in Aut G. Proof: If H FIK = H then G = K and we are done. Suppose H n K #’H. Then IH rlKl = p1 for some i such n—i that 0 3.1 < n. Thus H 0 K = (XP >. Let K = . Then G = . Let a €.Aut K. Since H rlK is characteristic in K, n-i a n-i a n—i L (xp ) is a generator of H rlK. Thus (xp ) = (xp ) where (£.p) = 1. If i = 0, we will take 1 = 1. Let g be an arbitrary element of G. Since G = HK, we can write 9 = xmk where 0 g_m < pn and k E‘K. Given 10 a and L as above, we define a map a from G to G by (ka)a = xm“ kg. The map a is well-defined and dh< = a. Claim: a is an automorphism of G. l and xsk2 be elements of G = HK. Since H‘S Z(G) and a E Aut K, (i) Let er r s a _ r+s 5 (x klx k2) - (x klkz) _ r£+s£ a — x (klkz) . xrzxsagkg _ r a s a _ r a s a — x Aklx lkz - (x k1) (x k2) . Hence a is a homomorphism. (ii) Since (£,p) = 1, x‘ generates H; and since a 6 Aut K. {Y?}§=1 generates K. Therefore a is onto. (iii) If 1 = (xrk)a = xr‘ko‘, then x” = (161)0‘ n-i n-i 6 H H K = . Thus xr‘ = (xp )m where i n-i n 1 g_m g.p . It follows that rt 5 m(p ) (mod p ). Since Llrz and (£,p) = 1, film. Let m = lq. . n—i Then r a qr-pn-l (mod pn) and xr‘ = (xp )q£ n-i -l = (xq°p )a = (xr)a. Thus xr = (xr‘)a = -1 ((k-l)a)a = km1 and xrk = 1. Therefore a is one-to-one. The mapping a 4 a maps Aut K to Aut G. Let a,B €.Aut K. Then dB 6 Aut K and 5,5,55, as defined above, are automorphisms of G such that alK = a. EIK = fl and 551K = 05. 11 First we show a a a is a homomorphism. Clearly y? = qu, j = 1,...,t. n-i n-i n-i Suppose (x p )S and (x where (s,p) = l = (r,p) and if H rlK 1, then r = s = 1. n-i n—i (xp )O‘B = ((xp )O‘)B n—i = <5 = (XP )rs. Clearly (rsapl = 1. Thus XOLE3 = xrs = (XE)S = (XS)E = (xa)E E—'= 55 on the generators so OBI: OS. If A = 5 then a = B. Thus the mapping a 4 5 embeds Aut K isomorphically in Aut G. Corollary 1.27. Let G be as in 1.26 and C g_K such that C is characteristic in G. Then C is characteristic in K. Proof: Let a E Aut K. Extend a to a 6 Aut G as above. Then Ca = Ca = C. Theorem 1.28. Let G be an extra-special p—group of exponent p, p odd. If H § G is an extra-special subgroup of G then H is not characteristic in G. Our proof depends upon the following results. 12 Lemma 1.29 ([4], p.195). Let C be an extra-special subgroup of the p—group P such that [P,C] g Z(C). Then P = CCP(C). In the following results, Hm denotes the central product of m cepies of the group H. Lemma 1.3Q_ ([4], p.204). An extra-special p—group P is the central product of r 2_l non—abelian subgroups of order p3. Moreover, if p is odd, P is isomorphic to Nk Mr"k where 2 N and P _ M o and [p] = 2r+1. From the proof of this lemma it follows that the r factors involved commute elementwise. Thus we have the following corollary. Corollary 1.31. Let P be an extra-special p—group, p odd, such that P is isomorphic to the central product of n 2_1 non-abelian subgroups Mj such that p p p M. = ,, , ,, , = ' . = , = = 1, 3 (X3 Y3|[XJ Y3] z X: Y) z [xj,z] = [yj,z] = 1) j = 1,...,n. 13 Then P = ., . ., . = , ., . = 1 ' ' ', . J 3 Proof of 1.28: By 1.30, H est Mr-k where N and M are as in the lemma and r'2 1. Since H 3’s and G has exponent p, exp H = p and thus k = 0. Since H is extra-special, we may choose x in H‘\Z(H) and y in H such that [x,y] = z ¢ 1. Thus (2) = Z(H) = Z(G) since H’ = Z(H) and G’ = Z(G) are cyclic of order p and H’ g_G’. Also, [G,G] g Z(G) = Z(H) so [G,H] g_Z(H). Lemma 1.29 implies G = HR where R = CG(H). Since R centralizes H and R is a p-group, Z(R) = Z(H). It follows that R is non-abelian for otherwise, G = H. Thus R’ = G’ = Z(G) and R/Z(G) g G/Z(G) is elementary MS R abelian so R is extra-special. Exp R = p. Thus R by 1.30, where s 2 1. Applying 1.31, H and R have the following presentations: H = and R = . 14 Since G = HK and K = CG(H), G= -: . -o - = o -o - = ' . 'o o Let t = r4-s. t Let g E G. Then we have 9 = ( H xj 0 c.,d.,a < , ° = 1,...,t. S" J J p 3 Define a mapping a:G o G by a t a c. a d. a g = < n (x.) 3 (y.) 3)z j=1 J 3 where a a . . . = ., . = . f lat (x ) X3 (Y3) Y3 1 3 # a _ a _ (X1) — xtl (Y1) — yt a a (Xt) ‘ Xl' (Yt) ’ Y1° The mapping a is a homomorphism which fixes Z(G) elementwise. Suppose g 6 Ker d. Then ga = l and hence cl d1 t-l c. d ct dt a J j _ (A) xt yt (j32 xj yj )x1 y1 z — l. c d t-l c. d. c d 1 1 j j t t yt (3'22 xj Yj )X1 Y1 left fixed by a. Therefore t It follows that x 6 Z(G) and is c d t-l c. d. c d t . 1 l j 3 t t _ 3 j Xt Yt (j=2 Xj Yj )X1 Y1 ' jfl xj Yj ' 15 Hence, by substituting this result into (A) we see that g = 1. Thus a is one—to-one. Since G is finite and a is one—to-one, a is an automorphism of G. By the way we have defined a it follows that Ha # H. Thus H is not characteristic in G. Since the original draft of this thesis it has been brought to our attention that Theorem 1.28 may be generalized as follows. If G is an extra-special p-group of exponent p then Z(G) is the only non-trivial characteristic subgroup of G. (See [5], Aufgabe 32, p.360). This generalization directly affects Theorem 2.3 and Theorem 1.32. Let G be an arbitrary group. Let H and K be characteristic subgroups of G such that H 2LK. Then there exists a homomorphism a 4 a mapping Aut G to Aut H/K such that a e cAutG (H/K) if and only if a = 1 in Aut(H/K). Here (xK)O‘ = xO‘K, x EH. Proof: Define 5:H/K 4 H/K by (xK)a = xO‘K x e H. The map a is well-defined for if xK = yK then xy-1 6 K. Since K is characteristic in G, xOL(yOL)'1 = O. 1)” eK. Thus xaK =yOK. Also, if x 61-1 then X en (xy- since H is characteristic in G and thus 3 maps H/K to H/K. 16 The map E is an automorphism of H/K. Furthermore, -1 a a E C (H/K) e x x 6 K for all x e H Aut(3 e xK = xaK for all x E H e xK = (xK)a for all x e H Let a,B E Aut G. Then OB 6 Aut G and E,§ and OB, as defined above, are automorphisms of H/K. Let x E_H. Then (xK)aB = xaBK = (xaK)B = ((xK)a B = (xK)aB. Hence a 4 5 is a homomorphism. Corollary 1.33. Let G,H,K and the map a 4 3’ be as in 1.32. Suppose [H:K] = p and a 6 Aut G has order a power of p. Then a 6 CAut(3(H/K)' Iggggf: Let a in Aut H/K be as defined in 1.32. Since a 4 5' is a homomorphism, [3| [a]. Also, since [H:K] = p, H/K is cyclic of order p and thus [Aut H/K] = p-1. Therefore [a] p-1. Since (p,p-1) = 1, a = 1 in Aut H/K. Hence by 1.32, a 6 CAut(3(H/K)° CHAPTER II p-GROUPS OF HALL TYPE In this chapter we show that every non-trivial closed normal stability group of a finite non-abelian p—group G of Hall type, p an odd prime, properly contains the group of inner automorphisms, Inn G, of G, provided G is not extra—special. Lemma 2.1. Let G be a finite non—abelian p—group of Hall type, p odd. Then (i) 01mm) = zmlmn. (ii) 6’ [01(G)J’= Z(01(G)) = “film” = elm) n Z(G). (iii) G/z(G) is elementary abelian. (iv) Ui(G) =ui(2(s)) for i 2 1. 01<2(o)) if [2(a)] >p 2(3) if [Z(G)] = p. (V) MG) = Proof: From result 1.20 (i), 01(G) is extra-special of exponent p and G = 01(G)Z(G). Let x = wxzx and y = wyzy be elements of G» where wk,wy E 01(G) and zx,zy 6 Z(G). l7 (i) (ii) (iii) (iV) (V) 18 Since G is a p—group, Z(G) #’l, whence 1 5" {21(Z(G)) _<__ Z(01(G)). Now 01(G) is extra—special and so [Z(Ql(G))l = p and it follows that 01(Z(G)) = Z(Ql(G)). For x,y 6 G, [x,y] = [wxzx,wy zy] = [wx,wy]. Thus G’.S [01(G)J’.S G’. Therefore G’ = [01(GH’. Also. [01(GH’ = zmlmn = “alum because 01(G) is extra-special. Hence by part (i), G’ = 01(Z(G)) and thus 1 7! G’ g Z(G) n 01(G) which is cyclic of exponent p. Therefore G' = 01(6) 0 Z(G). By part (ii), (3’ g Z(G). Therefore G/Z(G) is abelian. Moreover, since exp 01(G) = p and G = 01(G)Z(G), le(G)| = lwszZ(G)| = [wa(G)| = p for any x E G. Thus G/Z(G) is elementary abelian. It suffices to show thatzjl(G) =(31(Z(G)). For any x E G, xp = (wxzx)p = wfi zg = 2:. There- foreul(G) 301mm) 301(6). 1(6) . (See Since G is a p-group, §(G) = G’lj [5], p.272). Thus by part (iv), §(G) = G'Ul(Z(G)). If [Z(G)] = p then s' = 2(6) since by part (ii), 1 7’ G'_<_ Z(G) . Also, 01(Z(G)) =1. Thus MG) =c =Z(G). If [Z(G)] = pk where k > 1 then Z(G) = 19 where 2 has order pk since G is of Hall k—l k—2 type. But then G’ = = <(zp )1”) _<_ 01(z(G)). Thus <§(G)=01(Z(G)). In view of Lemma 2.1 (ii), we Obtain the following corollary of Theorem 1.26. Corollary 2.2. Let G be a finite non-abelian p—group of Hall type, p odd. Then Aut 01(G) can be embedded isomorphically in Aut G. 2522:: By Result 1.20 (i), G = Z(G)01(G). Since Z(G) is an abelian characteristic subgroup of G, it is cyclic of prime power order. By 2.1 (ii), Z(G) n 01(G) = [01(G)]' and thus is characteristic in 01(G). Set H = Z(G) and K = 01(G) and apply Theorem 1.26. Theorem 2.3. Let G be a finite non—abelian p-group of Hall type, p odd. If C is a non-abelian subgroup of 01(G) such that C is characteristic in G then C is extra-special. Proof: Since C is non-abelian, 1 # C’.g [01(G)]’. But |[Ql(G)]’| = p so C' = [01(G)]' is cyclic of order p. Furthermore, §(C) = C'01(C) = C' because C is a p-group and exp C = p. Since Z(C) is characteristic in C and C is characteristic in G, Z(C) is a characteristic abelian subgroup of G and thus is cyclic. Exp C = p. 20 whence [Z(C)] = p. This in turn implies Z(C) = C' since 1 # C’.§ C and Z(C) 2 Z(C) n.c’ # 1. Thus C is extra—special. Theorem 2.4. Let G be a finite non-abelian p-group of Hall type, p odd. Then the only subgroups of 01(G) which are characteristic in G are 01(G), 9(01(G)) and l. gooof: Let 1 # C's 01(6), where C is characteristic in G. Then 1 #’C n Z(G). Moreover, by 2.1 (ii), c n Z(G) _<_ 01(6) n Z(G) = “01“»). Thus, since ls(nl(s))l = p, c n Z(G) = “01“»). Hence i(ol(e)) gc. If C is abelian then C is a characteristic abelian subgroup of G and thus C is a cyclic subgroup of 01(G). It follows that [CI = p because exp C = p. We now have 1 # §(01(G)) S.C and [C] = p. Hence C = §(01(G)). Suppose C is non-abelian. Corollary 2.2 states that Aut 01(G) can be embedded isomorphically in Aut G. Thus C is characteristic in 01(G) as C.S 01(G) and C is characteristic in G. By 1.20 (i), 01(G) is extra—special of exponent p. Since C is a non-abelian subgroup of 01(6) and C is characteristic in G, Theorem 2.3 applies so C is extra-special. This in turn, by 1.28 implies that C is not characteristic in 01(G) unless C = 01(6). we conclude that if C is non-abelian then C = 01(6). Therefore, the only subgroups of 01(G) 'which are characteristic in G are l, §(01(G)) and 01(6). 21 Corollary 2.5. Let G be a finite non-abelian p— group of Hall type, p odd. If C is a critical subgroup of G then C = G. 2222:: Since C is critical, Z(G) g_cG(c) = Z(C) g,c. Let K = 01(G) n C # 1. Since K is the intersection of characteristic subgroups of G, K is characteristic in G. Hence by 2.4, either K = §(01(G)) or K = 01(G). If K = 9(01(G)) and c = wczc is any element of C where wC 6 01(6) and 2c 6 Z(G) g C then WC 6 K = 01(G) n C. Since K = §(01(G)), 2.1 (ii) implies K g Z(G) whence wC E Z(G) ‘which in turn implies c E Z(G). But then C = Z(G). This is a contradiction since Z(G) is not critical as CG(Z) = G and G'g Z(G) = Z(Z(G)). Hence K = 01(6) and G = Z(G)01(G) g c. Thus G = c. Remark 2.6. Essentially the same proof can be used to show that if C is a characteristic subgroup of G and Z(G) g c then either c = G or c = Z(G). Theorem 2.7. Let G be a finite non-abelian p—group of Hall type, p odd. If [Z(G)] = p‘ then 0k(G) = ukkm) (21(G), where L 2 k 2 1. Proof: Let Z(G) = be cyclic of order p‘. By 2.1 ‘”“T" . . j (iv). 03(6) =uj(z(G)). j_>_ 1. Clearly 03mm = . Let k.2 1. Let x = wxzx where wk 6 01(G) and 2x 6 Z(G). Then 22 x is a generator of Qk(G) k a (w 2 )P — 1 x x k t-k e (2X)p = 1 so zx 6 (2p > = Ubk(Z(G)). Thus if I, > k _>_ 1, then x is a generator of (3((G) L-k e x = zxwX for some 2x 6 U (Z(G)), w X 6 01(6) e x e (3"k(Z(G))01(G) e x e (gt-kw) 01(G) since l-k 2 l whence 2.1 (iv) implies 01—k(Z(G)) = Ubkm) . Hence 0km) = UL—k(G) 01(G) if L-k _>_ 1. I. L If k = 1., (3((G) = . Now xp = 1 e 14 l. I) (wxzx)p = l e (zx)p = 1. But ~[Z(G)] = pl whence (zx)p = 1 for any zx E Z(G). Thus 01(6) = G. Since oblm) 01(G) =UO(G)01(G) = G, we also have 01-k(G)01(G) = G: {2k(G) for k= 2. Theorem 2.8. Let G be a finite non—abelian p—group of Hall type, p odd. If [Z(G)] = pl then 0p(Aut G) = Stab(G > Q£_1(G) > 0‘4 (G) >...> 01(G) > G’ > 1). Proof: By Theorem 2.7, 0km) = 01—k(G) 01(G) for 12. 2 k > 1. Thus [0"]‘(G)|101(G)| l (6)] = _ Q, [u‘km n We)! pk|01(G)l p if _<_k < 1. [GI if k=£ 23 = kl|()1(G)| if l_<_k<£, 1191(6)] if k = z = pk-llol(G)l if 1 g k g 1. Therefore if 2 g_k 3.2 then I 0,, (G) l pk’ll 01 (6)1 [ (G): _ (6)] = = _ .- 0" Q" 1 “ix-1m] k 2101(GH Let a 6 Aut G have order a power of p. By Corollary 1.33, o e cAutG(nk(G)/nk_l(o)) for 2 gk 3 L and AutG(G ) since [0k(G): Qk_1(G)] = p and [G I Let s : G > “lo—1(6) > 0‘_2(G) >...> 01(6) > G’ > 1. L Stab(s) = fl cAutG(nk(G)/q(_1(s)) n CAutG(Ol(G)/G') n a E C k=2 C (G’) is a normal p—subgroup of Aut G since 5 is a Aut(3 characteristic chain and G is a p—group. (See 1.3 and 1.6). Thus Stab(s) 5-Op (Aut G). On the other hand, 0P(Aut G)‘g kQZ CAUt G (m(G) /nk-1(G)) n CAUtG (G ’) 0 Hence Stab (S) = 0p(Aut G) n C (01(G)/G' ). Let A = 0p(Aut G). By 1.17, Aut<3 A is a stability group. Therefore by 1.12, VGAi is characteristic in G for all i. Let j be the least integer such that YGAj 3 01(6). (we know j g_£-1 by 1.5 (i).) Since YGAj g_01(G) is a characteristic subgroup of G, Theorem 2.4 implies that YGAj = 91(G) or §(01(G)) or 1. If \(GAJ = 1 then A g Stab(s). If yGAJ a! 1 then VGAJ+1 < yGAj, 24 since there exists a smallest integer n(A) such that n(A) j+l yGA = 1. It follows that yGA = §(01(G)) or 1. Thus A _<_ CAutG Hence A = 0p(Aut G) g_Stab(s). (01(G)/G') since 9(01(G)) = G’ by 2.1 (ii). We conclude that 0P(Aut G) = Stab(s). In [1], Adney and Yen investigated CAut<3(G/Z(G)) which they called AC. They defined a purely non—abelian p-group to be a group which does not have an abelian direct factor and Obtained the following result. Result 2.9. If G is a p—group which is purely non— abelian then the group Ac of central automorphisms is also a p-group. Result 2.10 ([2], p.24). Let G be a p-group and Z be the center of G. If G"2 Z then Ac = Stab(G 2 z _>_ 1) genome/G22). Theorem 2.11. Let G be a non-abelian p—group of Hall type, p odd. Then G is purely non-abelian. Proof: Suppose not. Then G = A x B where A is abelian and A FIB = 1. Since G/B RsA and A is abelian, G' _<_ B. Thus G’ n A = 1. However, A _<_ Z(G), G’ g Z(G), [6’] = p and Z(G) is cyclic. Hence G" ,A. This is a contradiction. Therefore G is purely non-abelian. 25 Corollary 2.12. Let G be a non-abelian p—group of Hall type, p odd. Then the group AC of central automor- phisms is a p—group. Proof: The conclusion follows immediately from 2.9 and 2.11. Theorem 2.13. Let G be a non-abelian p-group of Hall type, p odd, such that [Z(G)] = pl. Let s :G 2 Z = 00(Z) 2 (31(2) 2...2 G' = (ft-1(2) > 1, where Z = Z(G). Then AC = Stab(s). Pgoof: Since Z(G) is cyclic of order pg, Ui(Z) is cyclic of order pL-i, 0 g_i 3.1. Therefore [01(2) :Ui+1(z)]= p for o_ i g z- 1. By 1.33, if a E Aut G has order a power of p then a 6 CAut(;QJi(Z)/ 1(2)) where 0 g_i g_£-l. Hence since AC is a p- group by 2.12, we have Stab(s) = Ac' Theorem 2.14. Let G be a non-abelian p-group of __ O Hall type, p odd. Then Ac — CAutGml (G)/G ). Proof: Let a E‘Ac and w E 01(G). Then wrlwa E Z(G) n 01(G) = G . Therefore a e CAutG (01(G)/G ). It follows that Ac 3 cAutG (01(G)/G ). Let a E C (01(G)/G') and g = E G ‘where Aut<3 W9 29 1 -1a z-l za - a W9 6 01(6) and 29 E Z(G). Then g ga 2g 2g wg wg 6 Z(G)- G = Z(G). Therefore a 6 AC and CAut(;(01(G)/G ) 5 Ac' Thus AC = CAut(;(01(G)/G ). 26 Corollary 2.15. Wlth G as above, CAut(3(Ql(G)/G ) is a p-group. Proof: The conclusion follows immediately from 2.12 and 2.14. Theorem 2.16. Let G be a non-abelian p-group of Hall type, p odd, such that [Z(G)] = pl. Then Ac = 0 Aut G . p( ) Proof: Let s : G > Q£_1(G) >...> 01(6) > G' > 1. Then by 2.8, 0p(Aut G) = Stab(s) and by 1.33 and 2.15, CAut(;(Ql(G)/G ) = Stab(s). Thus 2.14 implies 0p(Aut G) = Ac' Theorem 2.17. Let G be a non-abelian p-group of Hall type, p odd. Then for i,j 2 l, CAutG (G/UJ(G)) = CAut G (aim/ui+j (G)) n 0P(Aut s) . Proof: Let i,j 2.1. . j . . . . j Since 0 (G) is characteristic in G, CAut<3(G/U (G)) g Aut G. By 1.11 and 1.6, CAutG (G/UJ(G)) is a p-group since G is a p~group and 03(G) g §(G). Thus 3' CAut G (G/u (G)) g 0p(Aut G). Henceforth 03 ‘will denote 03(6) 'which by 2.1 (iv) equals 03(Z(G)). Since Z(G) is a characteristic abelian subgroup of G, it is cyclic. Let Z(G) = and suppose [x] = p‘. 27 . —1cx . If (1 écAutGm/UJ) then x x 603. Thus i -l i a —1 a i i+° (xp ) (xp ) = (x x )p 615 3. It follows that i i+' a ecAutGQ) /o 3). Now let a. e c G (Oi/(31+)) n 0p(Aut G). Theorem 2.8 Aut 1 implies a 6 C (01(G)/G’). Furthermore, (x-leL)p = Aut G i i C I O p —l p a 1+] -1 a j j (x ) (x ) E 0 so x x 6 (5 whence o. E CAut G (Z/o ). Ifl g = 29:79 6 GI where. 29 6 Z(G) and wg E ()1 (G) then - 0L _ - a - a j . = j J g g — 2g 2g wg W9 6 U G (5 . Hence a 6 CAutG (G/o ). j _ i i+j It follows that CAutG (G/o ) — CAutG (u A) ) n 0 (Aut G). P Corollary 2.18. If G is‘as in 2.17 and i+j = k then c (G/Uk'imn = c i AutG (G)) n 0P(Aut G). Aut G (0 Theorem 2.19. Let G be a finite non-abelian p-group i l of Hall type, p odd. Let A = Stab(s) where s : G >0 > 12 it. max[i1,j} 0 >--->U = 1 and I. 2 2. Then A = CAutG(G/U ) ik ik where j= max [is+1—is}. Here U =0 (G), s=1,....L-l k = 1,000; to Proof: Let t E [l,...,l-l] such that 1t+1-1t = 3. Then for any 5 E [l,...,L—l}, is+1 _g j+is and it follows i i i +j i i +j that u S >u “12 us . Thus CMtGHJ‘Q/us )g 1 1 5 3+1 CAutG(U /u ) for .s =.l,...,.¢-1, whence by 2.17, ° 13 13+1 cAutG (G/UJ) _<_ cAutG (u /u ) n Op(Aut G) for s = 1,..., 28 L—l i . i 3 3 3+ CAutG (G/u ) _g 891 CAut G (U /U 1) n Op(Aut G) i i t t+1 g cAut G (o /u ) n Op(Aut G) _ j . . _. _. — CAutG (G/U) by 2.18, Since 1t+l lt—j. i . _ 1 3 It follows that A - CAut G (G/U ) fl CAut G (6/!) ) CAut G (G/U )' Theorem 2.20. Let G be a finite non—abelian p—group j _ J' of Hall type, p odd. Then CAutG (G/U ) - CAut G (Z(G)/U ) n 0P(Aut G) for j 2 1 and uj = uj(G). Proof: Let j2 1. By 1.11 and 1.6, CAutG (G/uj) is a p—group since G is a p-group and (33 3 (MG). Thus CAutG (G/UJ) g j CAutG (Z(G)/0 ) n 0p(Aut G). Let oi e CAutG (Z(G)/uj) n 0P(Aut G). By Theorem 2.8, o. e CAutG(Ql(G)/G’). If g = zgwg e G where 29 e Z(G) and W9 6 01(G) then. g'lga = 2:512; wglw; 6 Uj . G' ___ Uj' Hence a 6 CAut G (G/UJ) . It follows that CAutG (G/uj) = CAutG (Z(G)/Uj) fl 0 AutG . p( ) Theorem 2.21. Let G be a finite non-abelian p-group of Hall type, p odd. Let A = Stab(s) where s : G > 11 12 it. Z(G) >0 >15 >...>u =1 and 1.22. Then A: / max{ilpj) { C (G U ) where j = max i AutG . . s=1,...,£-1 1k 1k Here (5 =0 (G) k=1,...,£. s+1 - ls} ° 29 Proof: As in 2.19, CAut(;(G/U ) = SCH CAut<3(U /U 3+1) n 0p(Aut G). Therefore i . _ 1 J = C i . 1 J 1 l _ 1 j - — CAut G (G/‘U ) n CAut G (G/o ), Since 0 max[il,j] AutCE Z(G) > 1). Thus A = Stab(G > [G,A] = G’ > [G,A,A] = 1) = Stab(G > G > 1) = c1!th (G/G) by 2.19. Let G = where Z(G) = (X) and 01(6) = . Since G is not extra—special, Z(G) > G’ thus if Ix] = pk then k.2 2. Define a mapping a from G to G by k-l X"XXP w. ... We. 1 1 Then as in the proof of 1.26, a is an automorphism of G. Furthermore, a e c (G/G’) \Inn G = A\A. Thus Inn G Aut<3 is not closed as a stability group. 30 Theorem 2.23. Let G be a finite non-abelian p-group of Hall type, p odd, G not extra-special. Let A be a non-trivial normal p-subgroup of Aut G. Then A = CAutG (G/UJ) for some j such that k - 1 _>_ j 2 l where [Z(G)] = pk. Proof: Since A is a normal p-subgroup of Aut G, A g 0p(Aut G). Thus by 2.14, and 2.16, A g c (01(G)/G'). Aut G Let g = zgwg 6 G and a €.A where 29 E Z(G) and -1 a _ -1 a -l a , _ wg 6 01(G). Then g g - Z9 29 W9 wg 6 Z(G) G — Z(G). Thus [G,A] g Z(G). By 1.17, 0P(Aut G) is a stability group, whence there exists an integer n such that V G 0p(Aut G)n = 1. This follows from 1.5 (1). Since A‘g 0P(Aut G), yGAn = 1 also. Now n #’l since A # 1. Furthermore, if [G,A] = Z(G) then n # 2, otherwise A = Stab(G > Z(G) > 1) and 1.25 implies A = Inn G since Z(G) is cyclic. But this is a contradiction since A is closed by 1.11 (iii) while Inn G is not closed (see 2.22). Therefore 1 i 1. A Stab(G>Z(G) >ul>...>u =1) 1 h/ N or 11 it Stab(G >0 >...>u = 1) L A 2. h/ . - - _ j In either case Theorems 2.19 and 2.21 imply A - CAut(3(GAU ) for some j such that 1 g_j g k-—1 where [Z(G)] = pk. 31 Corollary 2.24. Let G be as in 2.23. Let A be a non-trivial normal p-subgroup of Aut G. Then A = A if . _ j . and only if A - CAut(3(Gfl5 ) for some 3.2 1. Proof: The conclusion follows immediately from 2.23 and 1.13 since 0] g §(G). Theorem 2.25. Let G be a finite non-abelian p-group of Hall type, p odd, G not extra-special. If A is a non-trivial normal p—subgroup of Aut G then A > Inn G. . " _ 3' - Proof. By 2.23, A — CAut(3(Gfl5 ) for some 3 such that k- ‘2 jig l, ‘where [Z(G)] = pk. Since G’ = uk-1 3 U]. Inn G < A. Moreover, by 2.22, Inn G is not closed. Hence Inn G < A. Corollary 2.26. If G and A are as in 2.25 and A is closed as a stability group then A > Inn G. Proof: By 2.25 and the definition of closed, A = A > Inn G. Remark 2.27. Let G be an extra-special p—group of Hall type, p a prime. It follows from the proof of PrOposi— tion 3.2 in [7] that 0p(0ut G) = 1. Therefore Aut G possesses no normal p-subgroups properly containing Inn G. Moreover, 0P(Aut G) = Inn G) whence, by Corollary 1.17, Inn G is closed as a stability group. 32 An obvious question remains. What happens when G is a 2-group of Hall type, G not extra-special? Consider the Generalized Quaternion group of order 16 which is given by Q4 = and has the following subgroup lattice. 4 / \ 3 2 1 Since §(QG) = Q; = and Z(Q4) = , Q4 is not extra—special. MDreover, 04 is of Hall type. An automorphism of Q4 takes x to one of the four 3,x5,x7) and y to one of elements of order eight (x,x the eight elements of order four, (xly where i = 0,1,...,7). Thus [Aut 04] = 32. 33 Inn Q4 RzQ4/Z(Q4) = 04/ &:D where D is the Dihedral group of order eight. Set A = Stab(04'2 Z(Q4)'2 1). Then A is a normal abelian 2-group of Aut 04. Since Inn G RsD is non-abelian, Inn G g.A. The automorphism a such that xa = x5 and ya = y is a non—trivial automorphism in A. Moreover, A = A. Thus 1 #'A 42 Aut Q4 and A = A does not guarantee that A prOperly contains Inn G. CHAPTER III CLOSED SUBGROUPS AND LIFTING OF AUTOMORPHISM GROUPS As was demonstrated in Chapter I, an automorphism of a normal subgroup can sometimes be extended to an automorphism of the whole group. In this chapter we show that under certain conditions, lifting a stability group of a chain in a normal subgroup will produce a normal and even closed stability group in the automorphism group of the whole group. The material in this chapter stems from our consideration of p—groups which are not of Hall type and our efforts to determine the normal p—subgroups in the automorphism group of such a group. Let G be a non-abelian p—group, p a prime, and N o_G. We consider the group of extensions of N by G/N. We use the following results from cohomology groups: the notation Hn(B,A) denotes the nth cohomology group of B over A, n.2 1. (See [6], pp.128-130). Result 3.1 ([6], p.131). Let A be an abelian group. The second cohomology group H2(B,A) coincides with the group 34 35 of non-equivalent extensions of A by the group B with a given associated homomorphism 9. Result 3.2 ([6], p.142). Suppose there exist extensions of the group A by the group B with associated homomorphism 9. Then there exists a one-to—one correspondence between all non-equivalent extensions of A by B associated with e and all non-equivalent extensions of the center Z(A) of A by B associated with e. Schmid, in [7], Obtains the following criterion for extending automorphisms of normal subgroups to automorphisms of the whole group. Result 3.3 (Schmid, [7]). Let N'o G such that CG(N) = Z(N) and H2(G/N,Z(N)) = 1. Let D denote the group of automorphisms of N induced by G, and let AN. be any automorphism group of N. Then i) There is an automorphism group A of G normali- zing N and inducing AN on N if and only if AN normalizes D. ii) If AN centralizes D/Inn N, then AN can be lifted to G and any automorphism group of G extending A acts trivially on G/N. N Let G» be an arbitrary finite group. Suppose N‘g G and AN.S_Aut N such that AN can be lifted to A g_Aut G. The following question arises. What properties of AN are inherited by A? As a partial answer to this question when AN is a stability group, we have the following. 36 Theorem 3.4. Let N o_G where G is an arbitrary finite group. Let AN g_Stab(N = N0*2 N1'2"”2~Nk = 1). Suppose AN extends to a subgroup A of Aut G such that A centralizes G/N. Then YGAi _<_ YNAi-l _<_ Ni_1 for i 2 1. .ggoog: Induct on i. If i = l, YGA.g N since A centralizes G/N. Also, N = NG = YNAg = YNAO since A extends AN. Suppose YGAJ.S YNAJ'l S Nj-l' for j g.i. Then VGA1+1 = [YGA1,A] g [YNAl-1,A] = YNAl. Moreover, since A i-l . extends AN and yNAN '3 Ni—l' it follows that i-l _ i-l _ i—l ~ _ i [yNA ,A] — [yNAN ,A] — [YNAN ,AN] — YNAN 3 Ni. Let G,N,A and AN be as in 3.4. We have: Corollary 3.5. A g Stab(G 2 No 2 N1 _>_. ..__>_ Nk = l) and k A _<_ Stab(G 2 YNA: 2 YNA; 2...2 yNAN = 1). Let B = Stab(G.2lyNA§‘2 YNA;.2..22’YNA: = l) and BN = BIN. Then YNB; _<_ YNA; for all i 2 0. Since A g B, S-BN' Consequently YNA; g yNB; for all i'2 0. Thus ' i . l YNA; = YNBN for 1 2 0. Hence BN _<_ Stab (N 2 YNAN 2...2 k —— _ YNAN — 1) — AN and therefore AN'S-BN-S AN. If AN is closed then AN = BN and B is a stability group extending AN. If, in addition, N is characteristic in G and AN is normal in Aut N then by 1.3, B is 37 normal in Aut G since for i 2.0, YNA; is characteristic in N and thus in G. In summary, we have the following theorem. Theorem 3.6. Let N.o G where G is an arbitrary finite group. Let AN _<_ Stab (N = NO 2 N1 2...2 Nk = 1). Suppose AN extends to a subgroup A of Aut G such that A centralizes G/N. If AN is closed, then AN extends to a stability group in Aut G, namely B = Stab(G _>_ WA: _>_ YNAéIZ...2_yNA§ = 1). Moreover, if N is characteristic in G and AN is normal in Aut N then AN extends to a normal stability group in Aut G. Theorem 3.7. Let N.g G where G is a finite group. Let ANgAutN such that ANSStab(N=NGZN12...2Nk=1). Suppose A extends to a subgroup A of Aut G such that N 1 _ __. A g cAutG (G/YNA ). Set B — AIN. Then B 3 AN and B extends to A. Proog: Since N g_G, YNA g_yGA. 0n the other hand, YGA _<_ YNA since A g cAutG (G/YNA). Thus yGA = YNA which in turn implies YGAi = YNAi. It follows from 3.5 and 1.11 (ii) that for i 2 1, YNAi _<_ YGAi = YGAi = yNAi _<_ YNAi and so YNAi = YNAi. Let K‘N-S Aut N be the restriction of A to N. Then for i 2 o, YNAI; = mil, thus WA]; = YNAl = YNA;I for i 2 o. 38 It follows that AlN _<_ Stab(N 2 yNAN 2...2 WA: = 1) = Ag. Thus B = AIN g AE' and B extends to A. Corollary 3.8. Let G, AN’ A and B be as in 3.7. If AN is closed then A is closed. Proof: From 3.7, ANSAIN=BSA§. closed then B = AN’ and therefore A = A. Hence A is Thus if AN is closed. Remark 3.9. Let N be a normal subgroup of a finite group G such that CG(N) = Z(N) and H2(G/N,Z(N)) = 1. Suppose AN-S TN. such that AN centralizes D/Inn N where D is the group of automorphisms of N induced by G. Then AN = A;' in Aut N implies AN extends to a stability group in Aut G. Moreover, if N is characteristic in G and AN is normal in Aut N then AN extends to a normal stability group in Aut G. Proof: By 3.3, AN can be lifted to an automorphism group of G ‘which centralizes G/N. Now apply Theorem 3.6. Remark 3.10. It should be noted that: (i) If G is a p-group then a critical subgroup K of G is characteristic in G and has the property that CG(K) = Z(K). (See [4], p.185). Thus a proper critical subgroup of a p-group would be a candidate for N in 3.9. 39 (ii) If N is a prOper normal subgroup of a finite group G such that there is only one non- equivalent extension of N by G/N associated with a given homomorphism 9 then H2(G/N,Z(N)) 1. (See Results 3.1 and 3.2). The following results were motivated by my search for examples where the above theorems hold and my efforts to produce normal p—subgroups in the automorphism group of a finite p-group. Remark 3.11. Let G be a semi—direct product of K'g G and H. Let AK 3 Aut K. If the automorphisms of K in- duced by H are contained in CAutI((AK) then AK can be extended to a subgroup A of Aut G such that A g_CAut(3(G/K). Proof: Let a 6 AK. Define ma from (3 to G by T (hk) 0‘ = hko‘. In [2], Lemma 1.23, Bertelsen showed that ¢b' as defined above, is an automorphism of G = HK. Also a centralizes G/K. Let A = [male 6 AK]. A has the desired properties. Corollary 3.12. Let K be a finite group. If AK g_Aut K is an abelian group and G = K 1 AK then AK can be extended to a subgroup A of Aut G centralizing G/K. 40 Proof: Let a 6 AK' k 6 K. The automorphism which a for a induces on K is a itself since a-lka = k every k 62K. Thus the group of automorphisms of K induced by AK 15 just AK. Since AK [is abelian, AK S-CAutI<(AK)° Hence, by Remark 3.11, AK extends to a subgroup A of Aut G such that A.S_CAut(;(G/K). In the same vein, we have the following. Remark 3.13. Let K be a non-abelian p-group such that K’_<_Z(K). Let G=K p, and exp Z(K) = p. Let AK = Stab(K 2 Z(K) 2 1). By 1.25, with G =1K and N = AK>Inn K. Let 0L 6A.K\Inn K. Set G=K1. Then (i) CG(K) = Z(K) (ii) [G.K] 32(K) and (iii) cl(K) g_2 and K/Z(K) is elementary abelian. Proof: Let g = x8 6 G where x E K and l # B 6 Let k 6 K. (i) By 3.14, a has order p. Thus 8‘: a1 for some i, 0 < i < p. Also, since a A Inn K and [a] = p, it follows that G g Inn K. Now g-lkg = k @ B-lx-lka = k T e (k x)E5 k where mx 6 Inn K e (kB)cpX - k since Inn K g_ and ' Ax AK is abelian ()- e kB = k wk 4 B E Inn K. Thus CG(K) = Z(K). Z(K). (a). 42 (ii) Since K is a p—group, §(K) = K"-U](K). (See [5], p.272). Thus K’g (MK) _<_ Z(K). By 3.13, [K 1 AKfK] S_Z(K). Since [G.K] = [K :1 ,K] g [KaAK,K], we have [G,K] g Z(K). (iii) Result (iii) follows from the fact that MK) _<_Z(K). By result 1.8, abelian stability groups may be found by considering Stab(G'Z H.2 l) where H is an abelian subgroup of the group G. The following results produce closed abelian stability groups. Theorem 3.16. Let G be a non-abelian p-group. Let 1 #’H g_Z(G) such that exp H = p. Then B = Stab(G 2_H.2 l) is closed. 3mg: By 1.5 (i). [6.8] 3H and [G,B,B] = 1. Let 1 a! x 6 H and MR be a maximal subgroup of G containing Z(G). Then Mx-Q~G of index p and [x] = p. Hence there exists f e Hom(G,) with MK 3 ker f. Thus f e Hom(G/H,H). Let of E B be the corresponding automorphism of G guaranteed by 1.7. Then = [G,af] g [G,B]. Hence H _<_ [G,B]. Therefore H = [G,B] and B = Stab(G 2 [G,B] 2 l) = Stab(G2H2l) =B. 43 Corollary 3.17. Let G be a non-abelian p-group. Let 1 ¥ H‘g Z(G) and exp Z(G) = p. Then B = Stab(G 2LH‘2 1) is closed. Proof: Since H g_Z(G) and exp(Z(G)) = p, H has exponent p. Thus 3.10 applies to give the desired con- clusion. Corollary 3.18. Let G be a non-abelian p—group such that §(G) g_Z(G). Let H g_G’, the derived group. Then B = Stab(G 2 H'2 l) is closed. We first prove the following lemma. Lemma 3.19. Let G be a p-group such that G’Ig Z(G). Then exp G’ = exp(G/Z(G)). Proof: Since G’ g_Z(G), [x,y]3 = [x,y]] for j‘2 0 and x,y 6 G. If t = exp(G/Z(G)) then yt 6 Z(G) for every y 6 G. Thus [x,yt] = l for every x 6 G and hence [x,y]t 1 for every x,y 6 G. Thus exp G' g_exp(G/Z(G)). Now suppose t = exp G’ and x,y 6 G. Then [x,y]t = 1 = [x.yt] = 1 =’yt e Z(G) = exp(G/Z(G)) 3 exp G’o Hence exp G’ = exp(G/Z(G)). Proof of 3.18. Since i(G) g Z(G), exp(G/Z(G)) = p. Thus by 3.19, exp G’ = p. Also, since G is a p-group, 44 §(G) = G’Ul(G). (See [5], p.272). Thus G’.g Z(G). Since H.S G', it follows that exp H = p and H'g Z(G). There- fore by 3.16, B is closed. Theorem 3.20. Let G be a purely non—abelian p—group then the group AC of central automorphisms is closed. Proof: By results 2.9 and 2.10, AC is a normal p- subgroup of Aut G. Hence 0p(Aut G) is a stability group containing AC and we may consider AZ; Since YGAC.S Z(G), we have = Stab(G_>_ yGAC 2 YGA Definition 3.21. A p—group G is called special if either G is elementary abelian or G is of class 2 and I G = Z(G) = 9(G) is elementary abelian. Example 3.22. Let K be a special non—abelian p—group such that [Z(K)] > p then (i) K is purely non—abelian. (ii) Ac = Stab(K > Z(K) > 1) is a closed stability group which is elementary abelian and contains an outer automorphism a. (iii) Let a 6Ac be an outer automorphism. If G): K d , then Ac can be extended to a normal subgroup of Aut G which centralizes G/K and is closed as a stability group in Aut G. 45 ‘Proof: (1) Suppose K is not purely non-abelian. Then K = A x B, the direct product of A and B, where A is abelian. It follows that A g_Z(K). Also, A ng’ since K'.g B and A n B = 1. But this is a contradiction since K is special and therefore K’ = Z(K). Thus K is purely non-abelian. (ii) By 2.10, AC = Stab(K 2;Z(K)'2 1) since K’ = Z(K). Moreover part (i) and 3.20 imply that AC is closed. Since exp Z(K) = p, it follows from 3.14 that AC is elementary abelian. Finally, using Result 1.23, AC > Inn K since §(K) = Z(K) and Z(K) is not cyclic. (iii) Since a 6 Ac, Ac is abelian and the automorphism a induces on K, namely a-lka 4 kg for every k 6 K, is just a, Remark 3.11 states that AC can be extended to a subgroup A of G which centralizes G/K. Moreover, by Corollary 3.8, A is closed. BIBLIOGRAPHY BIBLIOGRAPHY Adney, J. and Yen, T., Automorphisms of a prgroup, Illinois Journal of Mathematics, 9 (1965), 137—143. Bertelsen, A.L., Some Stability Groups of Finite Groups, Ph.D. Thesis, Michigan State University, 1974. Gaschutz, W., Nichtabelsche p-Gruppen besitzen auBere p—Automorphismen, Journal of Algebra, 4 (1966), 1—2. Gorenstein, D., Finite Groups, Harper and Row, New York, 1968. Huppert, B., Endliche Gruppen I, Springer Verlag, Berlin, 1967. Kurosh, A.G., The Theory of Groups, Vol. II, Chelsea Publishing Company, New York, 1960. (Translated from Russian by K.A. Hirsch). Schmid, P., NOrmalgp-Subgroups in the Group of Outer Automorphisms of a Finite p-Group, Mathematische Zeitschrift, 147 (1976), 271—277. Schmid, P., Uber die Stabilitatsgruppen der Unter— gruppenreihen einer endlichen Gruppe, Mathematische Zeitschrift, 123 (1971), 318-324. Thompson, J., Normal p-complements for finite groops, Mathematicshe Zeitschrift, 72 (1960), 332-354. 46 M]I]I]]]][[][([]]I]]]]I[l[i[]l[l[