HWNIWIWH'Mll I {Hi 5 1 r WIIIIHIH N—I (0—: (DUMB 1 '—+ I 5555555555553 55%) 5555595553 T755333 5555 the fiegé‘ee an“ P5151 55555456553555 STATE UNEVERSETY P515535. 55555555653 55513525355? 33?} This is to certify that the thesis entitled M-groups and ill-groups presented by Paul Francis Murphy has been accepted towards fulfillment of the requirements for Ph . D degree in Mathematics Aloft/11X f [dill/£51 Major professor Dateégtmcfi 44;, /77/ 0-7639 g ., University imam! Michigan State w ABSTRACT M-Groups and 'IW-Groups By Paul Francis Murphy The purpose of this paper is to investigate group theoretic properties of M-groups. Interest in this area has been renewed by the work of Dade, Dornhoff and Seitz. The idea in this paper is to consider solvable groups which have no section of a certain type. The study of these types of sections is motivated by well known examples of groups which are not M-groups. A group G is a D-group if and only if: i) there are primes p, q, p + q, such that G is a (p,q) group. ii) G contains a normal non-abelian p-Sylow subgroup P such that Z(P) is cyclic and P/Z(P) is elementary abelian. iii) a Sylow q-subgroup of G has order q and acts trivially on Z(P) and non-trivially on P/Z(P). A group G is an E-group if and only if: i) for some prime , G has a non-abelian, normal p-subgroup P such that P/Z(Pg is elementary abelian and Z(P) is cyclic. ii) G acts irreducibly on P/Z(P) and Z(P) §_Z(G). A group G is an R-group if and only if G is an E-group, and G has a non-linear, faithful, primitive, irreducible representation. The main results of the paper are: i) Theorem 2.6: If G is a finite solvable group such that no section of G is a D-group, then G is an M-group. Paul Francis Murphy ii) Theorem 3.2: Let G be a finite solvable group. G is an -group (i.e. every section of G is an M-group) if and only if G has no section which is an R-group. The sufficient condition of Dornhoff and Huppert does not imply Theorem 2.6 and their condition is proved as a corollary to Theorem 3.2. Since Theorem 3.2 is clearly not a purely group-theoretic character- ization, the final section of the paper considers a possible improvement. M-GROUPS AND M-GROUPS By Paul Francis Murphy A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1971 Acknowledgements The author wishes to express his deep gratitude to Dr. David L. Winter for his encouragement and perserverence during the writing of this thesis. ii TABLE OF CONTENTS PAGE INTRODUCTION ......................... l CHAPTER I: Background From Representation Theory and Character Theory .............. 3 CHAPTER II: M-Groups .................... l2 3! CHAPTER III: M-Groups .................... 23 CHAPTER IV: Possible Improvement of Results ........ 27 BIBLIOGRAPHY ......................... 29 INTRODUCTION In this paper, all arbitrary groups are finite and all repre- sentations are over the field of complex numbers ¢. M-groups are characterized in the following representation theoretic manner: G is an M-group if and only if each irreducible representation of G is induced from a one-dimensional representation of a subgroup of G. The problem of proving a purely group-theoretic characterization of M-groups is unsolved. Vaguely, M-groups are (by set inclusion) between the class of solvable groups and the class of supersolvable groups. This thesis is concerned with the problem of learning more about the group theoretic properties of M-groups. In studying this problem, we are led to the investigation of a subclass of the class of M-groups. G is an’MZgroup if and only if every section of G is an M-group. The main idea of the thesis is that by requiring that a solvable group have no section of a certain type, we can then conclude that the group is an M-group. These certain types of sections are called D-groups and E-groups. They are motivated by examples in Chapter I of solvable groups which are not M-groups. The organization of this thesis is as follows. The first chapter and part of the second chapter are expository in nature and provide the reader with most of the background needed to understand the results of the paper, and the development of the problem. The remainder of Chapter 2 is the statement and proof of a group theoretic sufficient condition for a group to be an M-group; and an example to show that this condition is not implied by the sufficient condition of Huppert [10], and Dornhoff [4]. Chapter 3 contains a characterization of'MLgroups, from which we obtain the sufficient condition of Dornhoff as a corollary. This indicates that this characterization may be useful in proving further group theoretic sufficient conditions. Finally, because the characterization of M-groups is not purely group theoretic, Chapter 4 investigates the possibility of improving the results of Chapter 3. Chapter I Background from Representation Theory and Character Theory Definition: A representation T, of a group G, is a homomorphism of G into the group, GL(V), of nonsingular linear transformations of some finite dimensional vector space V over C. The dimension of V is the degree of the representation T, and V is called the representation space of T. Remark: For the purposes of this paper, we shall always assume that the ground field is the field C of complex numbers. Definition: A representation T of the group G, with representa- tion space V, is irreducible if there are no proper subspaces V1 of V, such that V]T(g) = V1 for all g e G. Definition: a) The character x, afforded by the representation T of G, is the function of G given by: x(9) = trace (T(g)) fbr g e G. b) The £2102. 01‘ x, ker x = {9 e G|T(g) = In} = {g e G|x(9) = n} where n is the degree of T. x is a faithful character if and only if ker X = l. c) x is irreducible if and only if T is irreducible. d) Degree of x = Degree of T; in notation deg x = deg T. e) A is a linear character if and only if A is afforded by a one dimensional representation. f) lG is the character afforded by the representation T that maps G onto 1 and 1G is called the principal character. 9) If x is a character of G, and H is a subgroup of G, le is the restriction of the function x to H. h) If x and ¢ are characters of a group G and x = o + w , where w is also a character of G, then o is called a constituent of x. If ¢ is irreducible, then o is an irreducible constituent of x. Definition: Let H be a subgroup of G and let G=l.) Hgi. If S i=l is a representation of H, we define the mapping SG of G, where SG(g)ij is the submatrix in the (i,j) position of 56(9): 5(959931) if (g, 9 95]) e H G - 5(g)lj- 1 0 ”(91-993)“ It can be shown that SG is a representation of G. 5G is called an induced representation. Definition: Let H be a subgroup of G and let G = L_) Hg If i=l ¢ is the character afforded by the representation S of H, then we 1-. . . . = w(g) if g e H . define the function 50 on G.wo(9) 0 if 9 ¢ H . Then the induced character 56, afforded by $6, is defined: wG(g) = ¢o(9i99i1) for u'vaa —l 1 g e G. Definition: A monomial matrix is a square matrix with exactly one non-zero entry in each row and column. A permutation matrix is a monomial matrix with the non-zero entries equal to l. Definition: A representation T of a group G is called a monomial (permutation) representation if T maps G into a group of monomial (permutation) matrices. Definition: Let T be a permutation representation of a group G and let V be the representation space of T. T(G) permutes the basis elements of V over C. If T(G) acts transitively on the basis of V, we say T is a transitive permutation representation. Remark: Let M be the group of all n x n monomial matrices over q, and let D be the subgroup of M consisting of all the diagonal matrices. Then, D d M, M = DSn and Df‘tSn =1, where Sn is the group of all n x n permutation matrices. See Ore [13]. If M1 is a subgroup of M and if n is the natural homomorphism of M onto Sn’ then we say n(M]) is the group of permutation matrices associated with M]. Thus, if T is a monomial representation of G of degree n, we call n°T the associated permutation representation of T, since nT(G) is a group of permutation matrices. Definition: A monomial representation T of the group G is transitive if the associated permutation representation of T is transi- tive. Definition: Two representations T and T', are equivalent if there exists a non-singular matrix S with entries in ¢ such that s"T(g) s = T'(g) for all g c c. The following is an important characterization of equivalent representations. (See Feit [5], page 12) Theorem l.l: Two representations T, T' of the group G which afford characters ¢, ¢' respectively are equivalent if and only if ¢ = ¢'. Definition: A group G is an M-group if and only if every irreducible representation of G is equivalent to a monomial representation. The next theorem and corollary yield a representation theoretic characterization of M-groups. (For proofs, see Scott [15] page 364.) Theorem l.2: A representation T of a group G is equivalent to a transitive monomial representation if and only if T is induced from a one-dimensional representation S of some subgroup H of G. Corollary 1.2.1: A group G is an M-group if and only if every irreducible representation T of G is induced from a one-dimensional representation S of some subgroup H of G. Corollary 1.2.1 is often used as the definition of M-group, as in Huppert [9]. Definition: If ¢ and w are characters of a group G, then we define the inner product ( , )G : (¢ , V06 = -T%J—- Z ¢(g)w(g). gEG Calculations using this inner product are very useful when work- ing with induced characters. Two results that indicate this are: 1) If x is an irreducible character of G, then (x,ip)G equals the number of times that x appears as an irreducible constituent of any character 5 of G. 2) Frobenius Reciprocity Law: If H is a subgroup of the group G and C is a character of H; x, a character of G, then (le. c)“ = (x . CG)G. The problem of finding a group theoretic characterization of M-groups is unsolved. In the next chapter, some of the difficulties that arise in working with the class of M-groups will be mentioned. Several mathematicians have proved sufficient conditions for a group G to be an M-group. The following condition is the only necessary condition for M-groups and gives us the only information that we have about an arbitrary M-group. Theorem 1.3: (Taketa [18]) Every M-group is solvable. Because of the importance of this theorem, we include its proof. First we extract two elementary but useful facts found in the proof and write them as lemmas. Lemma 1.3.1: If H is a non-trivial subgroup of G, then H is not contained in the kernel of some irreducible character of G. Proof: See Feit [5] for definition and properties of the right regular representation of G. Lemma 1.3.2: If A is a linear character of a subgroup H of the group G, then ker AG = Core (ker A) = /n\ y'](ker A)y yeG m Proof: Let G = L_) Hxi and let 9 e ker AG. i=l If S is a representation of H which affords A, then (56(9))ij = J J 5(x, gxt‘) = { 5 }; Iiikl . It follows that xigx;‘e ker A, for all i, l_: i_: m. Thus ge x;](ker A)xi. i=l Now let y be an arbitrary element in G. Since y = hxk for some k, l f-k 5_m, and since ker A.d H, y'](ker Aly = xglh'](ker A)h xk = x;](ker A)xk. Therefore €f§‘x§1(ker A)xi = /fi\ y'](ker A)y and g c Core(ker A). yeG 0n the other hand, if g e Core(ker A), 1 M3 fig) = x (x; o gxi) = m = AG(1) since (x;1 g xi) 5 ker A. l i Thus 9 e ker AG. Proof of Theorem 1.3: Let G be an M-group which is not solvable. Then 1 # G(i) = G(i+]) for some i. By Lemma 1.3.1, let T be an irreducible representation of minimal degree such that 6(1) £_ker T, and let x be the character afforded by T. Since G is an M-group, x = AG where A is a linear character of some subgroup H of G. Thus deg x = |G:H| . Let u = lH = lGI . By H . . G _ - - Frobenius Rec1procity, (16, u )G - (1”, “)H - 1. Therefore 1G 15 an irreducible constituent of ne. Since deg us = IG:H| , deg C < deg uG = deg x , for any irreducible constituent t of us. Hence 6(1) §_ker c, for all irreducible constituents C of us, so 6(1) §_ker us. By Lemma 1.3.2, ker uG = Core(ker u). Since u = 1”, Core(ker u) = Core(H) and 6(I) 5_H. Since G(i) = G(I+‘), e“) §_H'. Since H' is contained in the kernel of any linear character of H, 6(1) §_ker A. Finally, since G(124G and G(i) §_ker A, then G(i) §_Core (ker A) = ker AG = ker x. However this contradicts our choice of x. Solvability is certainly not a sufficient condition for a group G to be an M-group, as is illustrated by the following examples. The two examples are given in detail, since they will be used to motivate ideas in later chapters. Example 1.1: Let Q =(x,y|x4 = y4 =1, x2 = y2, y'1xy = {1) , the quaternion group of order 8, and let 0 e Aut Q = $4, with lol = 3. o is defined on Q: o(x) = y, 0(y) = xy, o(xy) = x. Now consider G = Q (q), the relative holomorph of G and (o) . The only elements of 0 fixed by «7) are l and x2. Thus x2 = 2(0) = Z(G). G/Q is abelian, so G' g Q. If IG‘I = 2, G' = Z(G) and G/Z(G) is abelian, so G is nilpotent, which implies G = Ox (0), a contradiction. If IG'I = 4, G' = (x) or‘(y> or (xy) . However, none of these groups is normal in G, since 0 permutes x, y, and xy, a contradiction. Therefore IG'I = 8 and G' = Q. Hence G has 3 linear characters. G/Z(G) contains no element of order 6, since 0' permutes x, y, and xy, so G/Z(G) ;IA4. Therefore G has irreducible characters of degree 1, 1, l, 3. Since the sum of the squares of the irreducible characters of G is 24, the degrees of the remaining characters must be 2. However G' = Q implies that G has no subgroup of index 2. Thus G is not an M-group. Before proceeding to Example 1.2, we introduce the following concepts. Definition: If H g_G and S is a representation of H which affords the character ¢ , then we define the following: a) For a fixed 9 e G, 59, the conjugate representation is defined: Sg(h) = S(g']hg) for any h e H. The conjugate character ¢g is the character afforded by 59. b) The inertial group of o , I(¢) is defined: 1(6) = {g e G |¢9 = o}. Since o is constant on conjugacy classes of H, H_<_I(¢)_<_G. The following results are due to Clifford [2], and are essential in working with induced representations. Theorem 1.4: Let H Q_G, and x be an irreducible character of G. Then XI” = k(¢ + ¢92 + "' + egt) where o is an irreducible character of H, and { egi, 1 §.i §_t } is a complete set of conjugate characters of o and k is a positive integer. Theorem 1.5: Let H g.G and let x be an irreducible character of G. Let ¢ be an irreducible constituent of le and let t be an irreducible constituent of x|1(¢) such that o 5'CIH . Then x = CG. Theorem 1.6: Let H fl_G and IG:H| = p, a prime. If 4 is an 10 irreducible character of H and I(¢) = G, then there is an irreducible character X of G such that XI” = 4- Finally, we use the following result about extraspecial p-groups. (Feit [5], page 128). 3 Theorem 1.7: If P is an extraspecial p-group of order p , then every non-linear character‘of P has degree p. Example 1.2: Let P be an extraspecial p-group of order p3 and exponent p, with p, an odd prime, such that p + l is divisible by an P=f=i.n¢]=na1=utmu=z> SL(2,p) is the group of outer automorphisms of P which fix Z(P), (Winter odd prime, q. P = (x.ysZ|XP = y [19]). Since |SL(2,p)| = p(p-1)(p+l), P has an automorphism,<5, of order q which fixes Z(P). (3 acts non-trivially on P/Z(P), an elementary abelian group of order p2, (Gorenstein [7], page 178). Every non-zero element of P/Z(P) generates a one dimensional subspace of P/Z(P), but p-l elements generate the same one dimensional subspace. Thus there are g:il'= p + 1 one dimensional subspaces of P/Z(P). SL(2,p) acts transitively on the one-dimensional subSpaces of P/Z(P). Let K be the subgroup of SL(2,p) which fixes a one dimensional subspace of P/Z(P). Since SL(2,p) acts transitively on the one- dimensional subSpaces, |SL(2,p):K| = p+l, the number of such subspaces. Since kII (p+l,<3 t K, i.e., 0 fixes no one dimensional subspace of P/Z(P). Now consider the group, G = P , |G| = p3q. Since |P'| = p, P has p2 linear characters. By Theorem 1.7, the remaining irreducible characters of G have degree p, and there are (p-l) such characters ll 3 = p20)2 + (p-l)(p)2. Let w be a non-linear irreducible since IPI = p character of P. Since P'd G, and |G:Pl = q, a prime, using Theorems 1.5 and 1.6, either a) $6 is irreducible, or b) XIP = W for some irreducible character X, of G. Suppose G is an M-group. If a) holds, G has a irreducible character of degree pq, namely 56. Since G is an M-group, there is a subgroup, H of G and a linear char- acter A of H such that AG = VG. Since IHI = p2, Z(P) = 2(6) f_H and IWG(z)| = I AG(z)| = pq for each 2 e Z(G). Therefore gZG IWG(g)| : P(PQ)2 > IGI. This contradicts the assumption that we is irreducible. If b) holds, then since G is an M-group, G contains a subgroup H of order p2 q, since deg X = p. If S is a Sylow p-subgroup of H, S 4 H since p >iq. Since (0) < H and S«q P, S 4 G. Therefore S/Z(P)Iq G/Z(P). Since S/Z(P) < P/Z(P), O fixes a one dimensional subspace of P/Z(P), a contradiction. Therefore G = P (0) is not an M-group. Chapter II M-Groups This chapter is partly expository and is designed to show how the results of this thesis fit in with the current knowledge of M-groups. The literature on M-groups is rather limited and new results have been infrequent. The recent work of Dade provides a partial explanation. (See Huppert [9], page 583 for proofs). Theorem 2.1: (Dade) If G is an M-group and C is a cyclic group of prime order, then the regular wreath product G O? C is an M-group. Corollary 2.1.1: (Dade) Every solvable group can be embedded in an M-group. Example 2.1: If G = Q (o) , as in Example 1.1 and ICI = IG/QI = 3, then G j_Q ’¥ C. (See Huppert [9], page 98). By Theorem 2.1, Q Q; C is an M-group which contains a subgroup Q‘(0) , which is not an M-group. Example 2.2: (Dornhoff [4]) The group in this example is also an M-group which contains Q‘(O> . Let Q, Q1 be two quaternion groups of order 8, and let QQ1 be their central product of order 32. Let 0 be an automorphism of order 3, such that Q0 = Q, Q? = Q], 0 non-trivial on both Q and 01; U is described in Example 1.1. Then the relative holomorph G = Q01 (0) is an M-group, and Q (o) _<_ G. The results of Dade give us an idea of the size of the class of M-groups, and further provide us with examples of M-groups, which have 12 l3 subgroups which are not M-groups. It is not known whether a normal subgroup of an M-group is an M-group, but there is the following theorem. Theorem 2.2: (Dornhoff [4]). If G is an M-group and H is a normal Hall subgroup of G, then H is an M-group. We now move to a consideration of sufficient conditions, and introduce the needed terminology. Definition: A section of a group G is a homomorphic image of a subgroup H of G. Definition: Let H2 be the quaternion group of order 8 and let Hp, for p an odd prime, be the extraspecial p-group of order p3 and exponent p. Then a group G is Hp-free if G has no section isomorphic to Hp. Definition: An irreducible representation T of a group G, with representation space V, is imprimitive if V = V1 9 V2 0 --- D Vr where Vi’ l 5.i 5_r, are subSpaces of V and r > 1, and each ViT(g) = Vj for l 5_j 5_r and g e G. An irreducible representation is primitive if it is not imprimitive. An irreducible character x is primitive if it is afforded by an irreduc- ible primitive representation. Definition: G is a primitive linear group, if and only if G has a faithful, primitive, non-linear, irreducible representation. Thus, for the purposes of this paper, all primitive linear groups are over C. Theorem 2.3: (Curtis-Reiner [3], pp. 346-48). Let T be an irreducible representation of a group G. T is primitive if and only if T is not induced from any irreducible representation S of a subgroup H of G. Definition: A group G is an A-group if G is solvable and every l4 Sylow p-subgroup of G is abelian for all p |G| . In the following theorem, Huppert generalized the sufficient conditions of [to [11], and Zassenhaus [21]. Theorem 2.4: (Huppert [10]). If a solvable group G has a normal subgroup N, such that G/N is supersolvable, and N is an A-group, then G is an M-group. In particular, all supersolvable groups, and all A-groups are M-groups. Theorem 2.4 is implied by the following theorem of Dornhoff. However, Seitz and Wright [17], showed that the two theorems are actually equivalent. Theorem 2.5: (Dornhoff [4]). If a solvable group G has a normal subgroup N such that G/N is supersolvable, and the Sylow p-subgroups of N are Hp-free for each prime p, then G is an M-group. He will present a proof of Theorem 2.5 in Chapter 3, as a corollary to another result. Now we present a collection of lemmas which are often used in this paper. Specifically, they will be used in the proof of the next theorem. Proofs of known results are presented when it is thought that the proof is different than existing proofs. For con- venience, we will call a faithful, primitive, non-linear irreducible representation (character) a restrained representation (character). Thus, for example, G is a primitive linear group if and only if G has a re- strained representation. Lemma 2.6.1: (Blichfeldt [1]). If G is a primitive linear group, then all normal abelian subgroups of G are contained in Z(G). Proof: Let H be a normal, abelian subgroup of G, and let ; be 9 9 a restrained character of G. By Theorem 1.4, CI“ = k(¢+¢ 2+oo~+¢ t). 15 If 1(9) 4 G, by Theorem 1.5, c = ,9 where w is an irreducible constituent of c|1(¢), such that ¢.§ wIH, a contradiction. So I(¢) = G. This implies :IH = ko. Since H is abelian, o is linear and c(l) = k. Therefore CIH = c(l)¢, and for all h e H, c(h) = C(l)w,w 2 ¢. If T is a representation of G that affords ;, T(h) is a scalar matrix for each h e H. Thus T(H) 5 Z(T(G)). Since T is faithful, H.: Z(G). Lemma 2.6.2: (Feit [5], page 46). If H §_K 5_G, and e is a K)G 2 as. character of H, then (6 Lemma 2.6.3: Let x be an irreducible character of the group G and let H < ker X < G with H 4 G. Let X' be the character of G/H G/H G associated with x(i.e. x'(Hg) = x(g)); if x' = A' , then X = A , (where A' is a character of some subgroup K/H of G/H and A is the associated character of K). 111 m Proof: Let G =tJ Kgi. Then G/H =LJ (K/H)Hgi. ““" i=l i=l G _ -l _ m ' -1 _ ' G/H A (g) - E Ao(gi 99%) -i§1Ao(Hgi 99,) — A 1 1 (H9). 5' G/H(H9) = x' = x 1. If (|M/Z(G)| ,|Z(G)|) = 1, M = AZ = A x 2, where A is an elementary abelian subgroup of M of order pr; this implies M is abelian, a contradiction. Therefore p l2(G)| . Let P be a Sylow p-subgrOUp of M, and let Z(G) = (PIA Z(G)) x 2]. Clearly (Pn Z(G)) _<_Z(P). |M| = p" |Z(G)|, |P| = p" |Pn Z(G)| and Ill = an Z(G)||Z]|. We have Z10 M and P21 _<_M. By the above, |PZ1| = |M| . Therefore M = P2 = P x Z and P char M 46, so P4 G. l 1 Hence Z(P) 4 G, and since Z(P) is abelian Z(P) §_Z(G), by Lemma 2.6.1. PZ(G)/Z(G) = '1; Hence Z(P) = Z(G)n P. Since M = PZ(G), P/Z(P) = P/Z(G)nP M/Z(G). Therefore P/Z(P) is elementary abelian. Hence P' f Z(P) and P' is cyclic since P' §_Z(P) 5_Z(G), which is cyclic by Lemma 2.6.4. Hence for x, y e P , xp 6 Z(P) and l = [xp,y] = [x,y]p. (See Huppert [9], page 253). Since P' is cyclic, | P'l = p. After considering Examples 1.1 and 1.2, we are led to the following class of groups. Definition: A group G is a D-group if it satisfies the following properties: 1) there are primes p, q, p + q such that G is a (p,q) group. 2) G contains a normal non-abelian p-Sylow subgroup P such that Z(P) is cyclic and P/Z(P) is elementary abelian. 3) A Sylow q-subgroup of G has order q and acts trivially on Z(P) and non-trivially on P/Z(P). 17 Definition: A group G is a Do-group if no section of G is a D-group. Lemma 2.6.6: Subgroups and factor groups of Do-groups are DO-groups. Eppgfz Obvious. Theorem 2.6: A solvable DO-group is an M-group. Epppf: Let G be a minimal counterexample and let C be an irre- ducible non-monomial character of G. By Lemma 2.6.6, every subgroup and factor group of G satisfy the hypothesis of the theorem. Therefore every proper subgroup and factor group of G is an M-group. Using this fact and Lemmas 2.6.2 and 2.6.3, C is a restrained character, and G is a primitive linear group. By Lemma 2.6.5, G has a normal, non-abelian subgroup P such that P/Z(P) is elementary abelian and Z(P) §_Z(G). Now let H be the centralizer of P/Z(P) in G. Suppose H + G. Then, there exists y E (G-H) such that Hy has prime order q. Suppose q + p. Consider (y) P/ (yq> . Notice that we can choose y so that pT lyl . If p lyl , i.e. lyl = pan, then since (Hy)q = H, (Hy)pa + H. Hence ypa 8 G-H and (H ypa)q = H. So we can use ypa if needed. Since Z(P) _<_ Z(G) and yq 6 H, yq stabilizes the normal series 1 4 Z(P)4 P. Since (I yq I, p) = l, yq centralizes P (see Gorenstein [7], page 178). Thus (yq> 4 (y) P and P/ H o _. 1 Z."' 3.Hi 2.“ m 1...: H" = CG(P) such thatIszuml e for D §_k §_n. For some i, O §.i §_n, CIH is irreducible and cl H is i i+1 reducible. 92 Qt By Theorem 1.4, CIH = k(¢+¢ +---+¢ ). If I(¢) + G, i+1 by Theorem 1.5, C = 49 where w is an irreducible character of I(¢) such that 4.: le , a contradiction. i+1 If I(¢) = G, then we consider the irreducible character cl” of Hi’ i and we see CIH = (CIH )IH . So we can consider the irreducible i+1 i i+1 character ClHi being restricted to the normal subgroup ”i+1 of index p in Hi‘ Since IG(¢) = G, IHi(¢) = Hi' By Theorem 1.6, (clHi)lHi+] = o, a contradiction since (C)lHi+] is reducible. The only remaining possibility is that G = H = CG(P/Z(P)). There- fore a Hall p'-subgroup K of G centralizes P/Z(P) and Z(P), so as before, K §_CG(P) and C/CG(P) is a p-group. We have just shown that this case leads to a contradiction, and the theorem is proved. It might be asked whether it is possible to extend Theorem 2.6 to a result of the form of Theorem 2.5 i.e.: If G is solvable and N54 G such that G/N is supersolvable and N is a DO-group, then G is an M-group. The answer is negative since 05(0>of Example 2.1 satisfies the hypothesis, but is not an M-group. 19 He now require some concepts from the theory of formations in order to describe the results of Seitz [16]. We shall apply these results as well as our Theorem 2.6 to an example. Definition: A class yof finite groups is a formation if: l) G e fimplies GO 8? for all epimorphic images G0 of G. 2) G/N], G/Nz c 3? implies G/N1n N2 cf. Definition: A formationEF‘is saturated if G/§(G) 6‘37 implies G e 3‘. Definition: For every prime p, let.§P(p) be a formation. A class ‘fof groups is called a locally defined formation, defined by €F(p), if‘j’consists of those groups G having the following properties: 1) pl |G| if?(p) = o 2) if H/K is a chief factor of G such that p I H/K I, then G/CG(H/K) e 1(p), where CG(H/K) = {o c Glth = hK for all h c H} = { g e GI [H, g ]e K }. 339935; If jFis a saturated formation of solvable groups and contains every nilpotent group, then G eTif and only if G/Z(G) e ‘37. This fact follows from the theorem of Lubeseder [12], that a saturated formation of solvable groups is locally defined. Classes of groups, X, where G 6 X if and only if G/Z(G) E X are used in the work of Seitz, and saturated formations which contain every nilpotent group provide examples of such classes. The classJ of supersolvable groups is a saturated formation which is locally defined as follows: for all p,¢{(p) consists of all abelian groups with exponent dividing p-l. (See Huppert [9], page 712, Hilfsatz 8.3). [lull'll:llel" 1"..1111'. 20 With this background, we now proceed to the sufficient condition of Seitz [16], which implies Theorem 2.5. Definition: LetC? be the union of all classes of solvable groups X such that X is subgroup closed, X is closed under homomorphisms, X consists of M-groups, G/Z(G) is in X if and only if G is in X. V Theorem 2.7: (Seitz [16]). Let G be a solvable group such that N g G and SIM eC . Suppose the p-Sylow subgroups of N are Hp-free for each prime p. Then G is an M-group. Corollary 2.7.1: (Seitz [16]). Lettfbe a saturated formation containing all nilpotent groups, and let'i’be subgroup closed and consist only of M-groups. If G/N is in HP, and the p-Sylow subgroups of N are Hp-free for all primes p, then G is an M-group. Let ‘33:! , the formation of supersolvable groups, and we have Theorem 2.5. In the process of obtaining this generalization, Seitz has introduced conditions which are not purely group theoretic; specifi- cally, the condition one that G e C implies that G/Z(G) is an M-group if and only if G is an M-group.r Proposition 2.8: (Seitz [16]). There exists a saturated form- ation ysuch that JG. 3:, fis subgroup closed,.¥contains every nilpotent group and ypcontainsfonly M-groups. The formation exhibited in this proposition is locally defined as follows: a) For p odd,.3r(P) is the class of abelian groups with exponent dividing p-l. b) .3’(2) is the class of elementary abelian 7-groups. 21 Theorem 2.9: (Dornhoff [4]). If P is a non—abelian 2-group with an automorphism a, of odd order n, acting without fixed points on P/iI>(P), P has a quaternion section. Theorem 2.9 will be used in the following example which gives groups which satisfy the hypothesis of Theorem 2.6 but not of Theorem 2.5. Definition: A Suzuki 2-group is a non-abelian 2-group, with more than one involution, having a cyclic group of automorphism which permutes its involutions transitively. 2" where Example 2.3: Let P be a Suzuki 2-group of order 2 (Zn-l) > 3 is a Mersenne prime. As described in Higman [8], P has the following properties: a) P has (Zn-1) involutions and P has an automorphism a which permutes the involution of P cyclically; lol = 2n-l b) Z(P) = P' = 910°) = HP); I Z(P) l = 2" c) P has exponent 4. Consider the relative holomorph, G = P <05) . Since a acts fixed-point-free on the involutions of P and exp P = 4, o acts fixed- point-free on P. (i.e., if p e P, lpl = 4 and o(p) = p, then 2 , a contradiction since p2 is an involution.) The property 04102) = p of being fixed-point-free is preserved under homomorphic images, so G acts fixed-point-free on P/§’(P). (See Gorenstein [7], page 335). By Theorem 2.9 P is not Hp-free. Also 1 <1 +(P)4 P4 G is a chief series for G, sincel §(P)| =l P/§(P)I = 2n and la] = 2n-l, a prime. Any pr0per subgroup P], normal in G, contains an involution and thus contains Z(P). So P is minimal in G with the property that GIP is supersolvable. Since P is not Hp-free, Theorem 2.5 does not apply. 22 Suppose G has a section H/K, which is a D—group. Then we can assume (a) .i H; so K is fixed by a . If K # 1, K contains an involution and Z(P) §_K. Since a acts cyclically on P/Z(P), K = Z(P) or K = P. But since H/K is a (p,q) group, K = Z(P) and H = G. However P (a) /Z(P) has an abelian p-Sylow group and is not a D-group. Thus K = 1. Thus H has a normal non-abelian p-Sylow subgroup P'such that Z(P) is cyclic and H acts trivially on 2(5). This implies a fixes the involution in Z(P), a contradiction. So G is a Do-group. Thus Theorem 2.6 implies that G. is an M-group. In the case n = 3, Seitz [16] shows that G belongs to the formation described in Proposition 2.8, and thus by Corollary 2.7.1, G is an M-group. Although it appears possible to exhibit a formation like that of Proposition 2.8 for the cases where (Zn-1) is any Mersenne prime, Theorem 2.6 is easier to apPIY- Chapter'III fl M-Groups p Definition: A group G is an M-group if and only if every section of G is an M-group. In the previous chapter, we considered the following type of theorem: If G satisfies hypothesis (H), then G is an M-group. For each such theorem, the following hold: 1) every subgroup S of G satisfies (H). 2) every homomorphic image, G/K, of G satisfies (H). l) and 2) imply that every homomorphic image of every subgroup of G satisfies H, so every section of G satisfies H. Therefore, since we proved G is an M-group if G satisfies (H), then every section of G is an M-group. Thus each sufficient condition in the previous chapter, is actually a sufficient condition for G to be an Negroup. In this chapter, we will prove the same type of theorem: If G satisfies (H), (where (H) is a hypothesis satisfied by all sections of G), then every section of G has property W. The following reasoning will be used. Let G be a minimal counter- example. Since every prOper section of G satisfies (H), every pr0per section has property W. Therefore only G itself does not have property W. Then, we will show that this leads to a contradiction. Theorem 3.1: G is an M-group if and only if no section of G has a restrained character. 23 24 fl 3599:; Certainly, if G is an M-group, every non-linear irreducible character of every section of G is induced from a linear character. 0n the other hand, let G be a minimal counterexample, i.e. no section of G has a restrained character and G is not an Migroup. Using the reasoning described above, G itself is not an M-group. There is a non-monomial irreducible character, ;, of G. Since every proper section of G is an M-group, we can use Lemmas 2.6.2 and 2.6.3 to show that g is a restrained character of G, a contradiction. Defintiion: A finite solvable group G is called an E-group if the following hold: 1) For some prime p, G has a non-abelian, normal, p-subgroup P such that P/Z(P) is elementary abelian. 2) G acts irreducibly on P/Z(P) and Z(P) 5_Z(G); Z(P) is cyclic. Definition: An E-group which has a restrained character is called an R-group. Definition: A solvable group G is called an Eo-group (respectively Ro-group) if and only if G has no section isomorphic to an E-group (respectively R-group). Theorem 3.2: Let G be a finite solvable group. G is an Negroup if and only if G is an Ro-group. Epppf: If G is an Hegroup, clearly G is an Ro-group by Theorem 3.1. On the other hand, let G be a minimal counterexample such that G is an RO-group, and G is not an M-group. Thus G itself is not an M-group. Let c be a non-monomial irreducible character of G. By the method of Theorem 3.1, g is a restrained character of G and G is a primitive linear group, and G has a normal non-abelian p-subgroup P, such that P/Z(P) is elementary abelian, and Z(P) 5 Z(G). Also Z(G) 25 contains all normal abelian subgroups of G. If G acts irreducibly on P/Z(P), then G is an R-group, a contra- diction. Thus G acts reducibly on P/Z(P) and irreducibly on some subgroup Pl/Z(P) 4 l of P/Z(P). Thus P1/Z(P) 4 G/Z(P), so P 4 G. 1 If P1 is abelian, P1 5_Z(G), and P1 5_Z(P), a contradiction. So PI/Z(P) is elementary abelian of order ps, 5 > 1. Z(P1)44 G implies Z(P1) 5_Z(G). Since Z(P) 5_Z(P]) 5_P] 5_P, Z(P) = Z(Pl). Thus G acts irreducibly on P1/Z(P]) where P1 is non-abelian, and Z(P1) 5_Z(G). Therefore G is an R-group, a contradiction. Theorem 3.3: (Rigby [14]). If G is a solvable primitive linear group, and P is a normal non-abelian p-subgroup in G, then P is not Hp-free. Using this result of Rigby, we now prove Theorem 2.5 as a corollary to Theorem 3.2. Corollary 3.2.1: If a solvable group G has a normal subgroup N such that G/N is supersolvable, and the Sylow p-subgroups of N are Hp-free for each prime p, then G is an Ro-group (i.e. an Nhgroup). 3599:: Let G be a minimal counterexample. By our previous discussion, and the fact that all proper sections of G satisfy the hypothesis, G is itself an R-group. Therefore, G has a normal, non- abelian subgroup P, such that Z(P) 5_Z(G) and G acts irreducibly on P/Z(P). Also G is a primitive linear group. Since (Prfi N) Z(P)‘4 G and G acts irreducibly on P/Z(P), (P n N) Z(P) = Z(P) or (PO N) Z(P) = P. Suppose (Pri N) Z(P) = P. Since P is non-abelian, Pri N is non-abelian. Since PIA N 0 G, PIA N is described in Theorem 3.3, 26 and Pfl N is not Hp-free. Since Pn N _<_ S , a Sylow p-subgroup of N, P Sp is not Hp-free, a contradiction. Therefore (PIA N) Z(P) = Z(P), so Prl N §_Z(P), and since [P,N] 5_Z(P), N acts trivially on P/Z(P). Therefore G/N acts irreducibly on P/Z(P). Also G/N acts irreducibly on K = [ PN/N / Z(P)N/N ]. If this is not the case, there is P], with Z(P) P, such that :Lpl 7i Z(P)N/N 4 PIN/N 4 PN/N d G/N is a normal series. Thus Z(P)N <1 P1N 0 PM 4 G is a normal series. Let p] e P]. Then for g c G g'lplg = p1*n 8 Pl"’ since PlN <1 G and g'lplg = p c P since P