finite groups having an involution centralizer with a 2 ... - Project Euclid

PACIFIC JOURNAL OF MATHEMATICS Vol. 87, No. 1, 1980

FINITE GROUPS HAVING AN INVOLUTION CENTRALIZER WITH A 2-COMPONENT OF TYPE PSL (3, 3) MORTON E.

HARRIS

A finite group L is said to be quasisimple if L—V and L/Z(L) is simple and is said to be 2-quasisimple if L— ΊJ and L/O(L) is quasisimple. Let G denote a finite group. Then E(G) is the subgroup of G generated by all subnormal quasisimple subgroups of G and F*(G)—E(G)F(G) where F(G) is the Fitting subgroup of G. Also a subnormal quasisimple subgroup of G is called a component of G and a subnormal 2-quasisimple subgroup of G is called a 2-component of G.

We can now state the main result of this paper: THEOREM A. Let G be a finite group with F*(G) simple. Assume that G contains an involution t such that H=CG{t) possesses a ^-component L with L/O(L) = PSL (3, 3) and such that CH(L/O(L)) has cyclic Sylow 2-subgroups. Then \F*(G)\2 ^ 210.

In order to state an important consequence of Theorem A, we require two more definitions. A subgroup K of a finite group G is said to be tightly embedded (in G) if \K\ is even and \K(\Kg\ is odd for every g eG — NG(K). A quasisimple subgroup L of a finite group G is said to be standard (in G) if [L, L9] Φ 1 for all geG, CG{L) is tightly embedded in G and NG(L) = NG(CG(L)). THEOREM B. Let G be a finite group with O(G) = 1 and containing a standard subgroup L with L = PSL(3, 3). Then either L^QG or L Φ (LGy = JP*(G) and one of the following five conditions hold: (a) F*(G) ~ PSL (3, 9); (b) F*{G) = PSL (4, 3); (c) F*(G) = PSL (5, 3); (d) i^*(G)^PSp(6,3); (e) F*(G) - H,x H2 with H^H^L and CG(L) = (t) where t is an involution such that HI — H2 and L = (hjn\ \ ΛX6 flj).

Note that Theorem B is a step toward the verification of Hypothesis θ* of [13] and is therefore of import for completing a proof of the Unbalanced Group Conjecture and the J?((τ)-Conjecture and for completing an inductive characterization of all Chevalley groups over finite fields of characteristic 3 (cf. [13, § 1]). Also by applying 69

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[13, Lemma 2.9], [3, Theorem], [1, Corollary II], [8, Theorem 5.4.10 (ii)|, [3, Table 1] and [6, Tables 3 and 4], it suffices, in proving Theorem B, to assume, in addition to O(G) = 1, that L Φ F*(G) = G (L ), F*(G) is simple and that CG(L) has cyclic Sylow 2-subgroups. But then Theorem A and the classification of all finite simple groups 10 whose Sylow 2-subgroups have order dividing 2 (cf. [4] and [7]) yield Theorem B. Consequently Theorem B is a consequence of Theorem A. The remainder of this paper is devoted to demonstrating that the analysis of [12] and [14] can be applied to prove Theorem A. All groups in this paper are finite. Our notation is standard and tends to follow the notation of [8], [12] and [14]. In particular, if X is a (finite) group, then S(X) denotes the solvable radical of X, O2(X) is the subgroup of X generated by all elements of X of odd order and is consequently the intersection of all normal subgroups 7 of I such that X/Y is a 2-group and &(X) denotes the set of elementary abelian 2-subgroups of X. Also, if n is a positive integer, then gfw(X) denotes the set of elementary abelian 2-subgroups of order n of X. Finally m2(X) denotes the maximal rank of the elements of g'(X), r2(X) denotes the minimal integer k such that every 2-subgroup of X can be generated by k elements and if Y £ X, then J"'( Y) denotes the set of involutions contained in Y. Clearly, if X is a group, then m 2 (X)^r 2 (X) and r 2 (X)^r 2 (Γ) + r2(X/Y) for every normal subgroup Y of X. 2* A proof of Theorem A* Throughout the remainder of this paper, we shall let G, t, H and L be as in the hypotheses of Theorem A and we shall assume that \F*(G)\2 > 210. Then [9, Main Theorem], [15, Four Generator Theorem], [3, Table 1], [6, Tables 3 and 4] and [2] imply that 42 and S < N^S). N S) Since Z(S)^NG(S) and ^ iSΓG(S), we have \NAS)/S\=2f t ^ ^= t{z) and Z(NAS)) = = Z{S?). Clearly O(CG(S)) = O(NG(S)) x and if π is an element of odd order of NG(S)f then π 6 CG«ί, «>), π e CG(X), π 6 CG«ίf u> x D) and hence πeO(NG(S)). Thus iSΓG(S) - O(NG(S))NAS). As in [12, § 4], we have SCNδ(S^) = φ and there is an element Eeϊ?B(^) such that E^S". Clearly «6 j&, | C^ί) | ^ 4 and ze CE(t) ^ S = C^(ί). Suppose that τetGf)E. Then | £f \ = | τ ^ | | C^ (τ)| ^ 22 I S| = 28. Thus tGf}E = φ,tZ CE(t\ \ CE(t) \ = 4, = X = , [ S , ί ? ] ^ £ / n S = (?,(«), iV^(S) = SE and ί* = t(z). Interchanging % and tu if necessary, it follows that we may assume that CEit) = . Set F = . Then A = ί 7 U ί ί 7 U uF U toi77, tF^tG Π A, tG n 7 (ί U wJF) = 0 and ί F £ tG Π A £ ί F U tuF. Consequently: 2.6. Either tM_= tG f) A = tF =:tGΓ)A = tFΌtuF and \M/A\ = 48. COROLLARY

tM

and \M/A\ = 24 or

Now the analyses of [12, § 5-11], with the obvious slight changes, shows that |O2(G)|2 ^ 210. Since F*(G) ^O2(G), our proof of Theorem A is complete. REFERENCES 1. M. Aschbacher, A characterization of Chevalley groups over fields of odd order, Ann. of Math., 106 (1977), 353-398. 2. M. Aschbacher and G. M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Math. J., 6 3 (1976), 1-91. 3. , On groups with a standard component of known type, Osaka J. Math., 13 (1976), 439-482.M 4. B. Beisiegel, Uber einfache gruppen mit Sylow 2-gruppen der ordnung h'όchstens 210, Comm. in Alg., 5 (1977), 113-170. 5. H. Bender, On groups with Abelian Sylow 2-subgroups, Math. Z., 117 (1970), 164-176. 6. N. Burgoyne and C. Williamson, Centralizers of involutions in Chevalley groups of odd characteristic, unpublished dittoed notes. 7. F. J. Fritz, On centralizers of involutions with components of 2-rank two, I, II, J. Alg., 4 7 (1977), 323-399.

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8. D. Gorenstein, Finite Groups, Harper and Row, New York, 1968. 9. D. Gorenstein and K. Harada, Finite groups whose 2-subgroups are generated by at most A-elements, Mem. Amer. Math. Soc. No. 147, Amer, Math. Soc, Providence, R. I., 1974. 10. D. Gorenstein and J. H. Walter, Balance and generation in finite groups, J. Alg., 3 3 (1975), 224-287. 11. K. Harada, On finite groups having self-centralizing 2-subgroups of small order, J. Alg., 33 (1975), 144-160. 12. M. E. Harris, Finite groups having on involution centralizer with a 2-component of dihedral type, II, Illinois J. Math., 2 1 (1977), 621-647. 13. M. E. Harris, On PSL (2, q)-type 2-components and the unbalanced group conjecture, to appear. 14. M. E. Harris and R. Solomon, Finite groups having an involution centralizer with a 2-component of dihedral type, I, Illinois J. Math., 2 1 (1977), 575-620. 15. A. MacWilliams, On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups, Trans. Amer. Math. Soc, 150 (1970), 345-408. Received June 21, 1978. This research was partially supported by a National Science Foundation Grant. UNIVERSITY OP MINNESOTA

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