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Chapter Iv, from Solvability of Groups of Odd Order, Pacific J. Math., Vol. 13, No Pacific Journal of Mathematics CHAPTER IV, FROM SOLVABILITY OF GROUPS OF ODD ORDER, PACIFIC J. MATH., VOL. 13, NO. 3 (1963 WALTER FEIT AND JOHN GRIGGS THOMPSON Vol. 13, No. 3 May 1963 CHAPTER IV 14* Statement of Results Proved in Chapter IV In this chapter, we begin the proof of the main theorem of this paper. The proof is by contradiction. If the theorem is false, a minimal counterexample is seen to be a non cyclic simple group all of whose proper subgroups are solvable. Such a group is called a minimal simple group. Throughout the remainder of this chapter, © is a minimal simple group of odd order. We will eventually derive a contradiction from the assumed existence of ©. In this section, the results to be proved in this chapter are summar- ized. Several definitions are required. Let 7T* be the subset of n(®) consisting of all primes p such that if sp is a Sp-subgroup of ®, then either Sfif^ift) is empty or $P contains a subgroup 21 of order p such that C^(2l) = 21 x 95 where 33 is cyclic. Let n? be the subset of TC* consisting of those p such that if ^5 is a Sp-subgroup of © and a is the order of a cyclic subgroup of iV($P)/$pC($P), then one of the following possibilities occurs: (i) a divides p — 1. (ii) $P is abelian and a divides p + 1. (iii) | $P | = pz and a divides p + 1. We now define five types of subgroups of ©. The basic property shared by these five types is that they are all maximal subgroups of ©. Thus, for x = 1, II, III, IV, V, any group of type x is by definition a maximal subgroup of @. The remaining properties are more detailed. We say that 9K is of type I provided (i) 3K is of Frobenius type with Frobenius kernel §. (ii) One of the following conditions is satisfied: (a) § is a T. I. set in ©. (b) 7r(£)S7r*. (c) & is abelian and m(§) = 2. (iii) If p e 7r(3K/§), then mp(2TC) ^ 2 and a SP-subgroup of 2Ji is abelian. The remaining four types are by definition three step groups. If @ is a three step group, we use the following notation: @ = ©'2^ , ©' n 25*! = 1 , C&iZ&J = SB*. *•• • • Furthermore, !Q denotes the maximal normal niljxrtent S-subgroup of ©. By definition, §^S' so WG let II be a complement for § in ©', 845 846 SOLVABILITY OF GROUPS OF ODD ORDER In addition to being a three step group, each of the remaining four types has the property that if 2Q0 is any non empty subset of aO^aOS, - SB1 - 2B2, then iV@(2B0) = SJyBB,, by definition. The remaining properties are more detailed. We say that @ is of type II provided (i) U =£ 1 and U is abelian. (ii) N9(O)&&. (iii) iV®(2I) g @ for every non empty subset SI of @" such that (iv) | 2&! | is a prime. (v) For every prime p, if 2t0, 2tx are cyclic p-subgroups of U which are conjugate in © but are not conjugate in @, then either C^(2I0) = 1 or Cc(8y = 1. (vi) £>C(£>) is a T. I. set in @. We say that @ is of type III provided (ii) in the preceding defi- nition is replaced by (ii)' Ai(U)S©f and the remaining conditions hold. We say that @ is of type IV provided (i) and (ii) in the definition of type II are replaced by (i)" U'*l, (ii)" AT<,(U)S©f and the remaining conditions hold. We say that <S is of type V provided (i) U = l. (ii) One of the following statements is true: (a) @' is a T. I. set in ©. (b) @' = sp x @Of where @0 is cyclic and ty is a Sp-subgroup of © with THEOREM 14.1. Let © 6e a minimal simple group of odd order. Two elements of a nilpotent S-subgroup § of © are conjugate in © i/ and only if they are conjugate in iV(£>). Either (i) or (ii) is true: (i) Every maximal subgroup of © is of type I. (ii) (a) © contains a cyclic subgroup 23 = SBx x 2B2 wi</& £fte property that JV(2B0)=2B /or ever]/ non empty sw&set 2B0 o/ SB—2$^-SB!. AZso, SB, * 1, i = 1, 2. (b) © contains maximal subgroups @ and 2 not o/ type 7 e 14. STATEMENT OF RESULTS PROVED IN CHAPTER IV 847 (c) Every maximal subgroup of ® is either conjugate to @ or X or is of type I. (d) Either @ or Z is of type II. (e) Both @ and % are of type II, III, IV, or V. (They are not necessarily of the same type.) In order to state the next theorem we need further notation. If 8 is of type I, let 8 = 2, = U C2(H) , where £> is the Frobenius kernel of 8. If 8 is of type II, III, IV, or V, we write 8 = 8'SB^ 8' n SBi = 1. Let £> be the maximal normal nilpotent S-subgroup of 8, let 11 be a complement for § in 8' and set SB = C^SBx), 2Ba = 2B n 8', 3& = 2B - If 8 is of type II, let 8 = U C2,(H) . ne§* If 8 is of type III, IV, or V, let 8 = 8'. If 8 is of type II, III, IV, or V, let 8t = 8 U U L- zest We next define a set s/ = J^(8) of subgroups associated to 8. Namely, 3Ji e sf if and only if 2JI is a maximal subgroup of © and there is an element L in 8* such that C(L) g; 8 and C(L) g; 9Ji. Let {??!, • • •, SJJJ be a subset of s*/ which is maximal with the property that 9^ and Sfty are not conjugate if i =£ i. For 1 ^ i g w, let £>* be the maximal normal nilpotent S-subgroup of %. THEOREM 14.2. If 8 is of type 7, //, ///, IV, or V, then 8 and 8X are tamely imbedded subsets of © with JV(S) = N(k) = 8 . JTf J^(8) is empty, 8 and 8X are T. I. sets in ©. // j^(8) is non empty, the subgroups $lf • • •, &n are a system of supporting subgroups for 8 and for 8^ The purpose of Chapter IV is to provide proofs for these two theorems. 848 SOLVABILITY OF GROUPS OF ODD ORDER 15. A Partition of 7r(©) We partition 7r(©) into four subsets, some of which may be empty: n1 = {p\A Sp-subgroup of © is a non identity cyclic group.} 7T, = {p 11. A Sp-subgroup of © is non cyclic. 2. © does not contain an elementary subgroup of order p3.} 3 7T8 = {p 11. © contains an elementary subgroup of order p . 2. If ^P is a Sp-subgroup of ©, then M(^P) contains a non identity subgroup.} 3 7r4 = {p 11. © contains an elementary subgroup of order p . 2. If ^P is a Sp-subgroup of ©, then M(^P) contains only It is immediate that the sets partition 7r(®). The purpose of Lemma 8.4 (i) is that condition 2 defining 7ra is equivalent to the statement that Sf^^K^) is empty if ty is a Sp-subgroup of ©. Lemma 8.5 implies that 3 g nx U TT2. 16. Lemmas about Commutators Following P. Hall [19], we adopt the notation 72133 = [21,S3], 2 r+i2ls8»+i = [r.gsB«f »], n = lf 2f • • •f and 7 2I33(S: = [21, 93, £]. If X is a group, ^f3^(X) denotes the set of normal abelian subgroups of X. The following lemmas parallel Lemma 5.6 of [27] and in the presence of (B) absorb much of the difficulty of the proof of Theorem 14.1. LEMMA 16.1. Let ^ be a Sp-subgroup of © and 2t an element of ^4C^(^). If % is a subgroup of © such that (i) <2t, S> is a p-group, (ii) % centralizes some element of Z(ty) fl 21*, then 73S2t3 = Proof. Let Ze C(g) n Z(¥) n 21*, and let G = C(Z). By Lemma 7.2 (1) we have 21 S <W<£) = ©. As sp is a Sp-subgroup of <£f Sft = ^P 0 § is a Sp-subgroup of §. Since 21 < spf so also 21 < SR, and since 2 21 is abelian, we see that 7 £>2P £ OP>(£). Since § < £, we have 3 7^21 S § and so 7 g2F S OP,(C). Since <2l, g> is assumed to be a p- group, the lemma follows. If ^P is a non cyclic p-group, we define ^<0P) as follows: in case is non cyclic, ^(^P) consists of all subgroups of Z(^S) of type (p, p); in case Z(^P) is cyclic, ^(^P) consists of all normal abelian subgroups of ^P of type (p, p). LEMMA 16.2. Let tybea non cyclic SPsubgroup of ©, 21 e 16. LEMMAS ABOUT COMMUTATORS 849 and let % be a subgroiip such that » j :r (1) <% 3> is a p-group, (ii) 21 contains a subgroup S3 of %/(%) such that // p ^ 5, then 74S2l4 = <1>, while if p = 3, tAen 76g2lQ =; <1>. t/ Hi == 51 n Z(«P) and p^5, then 7332t? = Proo/. If 33O £ Z(SP), the lemma follows from Lemma 16.1. If S30 g Z(?P)f then $P0 = C^(a30) is of index p in ^P so is of index at most p in a suitable Sp-subgroup ^p* of C(33O) = G. In particular, ^po < *P*. Let § = (V,(G)f SRf = $* n ft, and % = *po n ft. Since % < *P*, 3 3 so also «R < *P*. Hence 7?fc*H S % n ft S «Rf and so 7 W2I = <1>, 21 being in ^K^f(%).
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