Bender groups as standard subgroups

Griess, Robert L.; Mason, David R.; Seitz, Gary M.

doi:10.1090/S0002-9947-1978-0466300-7

Bender groups as standard subgroups
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by Robert L. Griess, David R. Mason and Gary M. Seitz PDF

Trans. Amer. Math. Soc. 238 (1978), 179-211 Request permission

Abstract:

A subgroup X of a finite group G is called $^ \ast$-standard if $\tilde X = X/O(X)$ is quasisimple, $Y = {C_G}(X)$ is tightly embedded in G and ${N_G}(X) = {N_G}(Y)$. This generalizes the notion of standard subgroups. Theorem. Let G be a finite group with $O(G) = 1$. Suppose X is $^ \ast$-standard in G and $\tilde X/Z(\tilde X) \cong {L_2}({2^n}),{U_3}({2^n})$ or ${\text {Sz}}({2^n})$. Assume $X \ntriangleleft G$. Then $O(X) = 1$ and one of the following holds: $({\text {i}})\;E(G) \cong X \times X$. $({\text {ii}})\;X \cong {L_2}({2^n})$ and $E(G) \cong {L_2}({2^{2n}}),{U_3}({2^n})\;or\;{L_3}({2^n})$. $({\text {iii}})\;X \cong {U_3}({2^n})$ and $E(G) \cong {L_3}({2^{2n}})$. $({\text {iv}})\;X \cong {\text {Sz}}({2^n})$ and $E(G) \cong {\text {Sp}}(4,{2^n})$. $({\text {v}})\;X \cong {L_2}(4)$ and $E(G) \cong {M_{12}},{A_9},{J_1},{J_2},{A_7},{L_2}(25),{L_3}(5)\;or\;{U_3}(5)$. $({\text {vi}})\;X \cong {\text {Sz}}(8)$ and $E(G) \cong {\text {Ru}}$ (the Rudvalis group). $({\text {vii}})\;X \cong {L_2}(8)$ and $E(G) \cong {G_2}(3)$. $({\text {viii}})\;X \cong {\text {SL}}(2,5)$ and G has sectional 2-rank at most 4. In particular, if G is simple, $G \cong {M_{12}},{A_9},{J_1},{J_2},{\text {Ru}},{U_3}(5),{L_3}(5),{G_2}(5), or\;{^3}{D_4}(5)$.

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