Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Bender groups as standard subgroups


Authors: Robert L. Griess, David R. Mason and Gary M. Seitz
Journal: Trans. Amer. Math. Soc. 238 (1978), 179-211
MSC: Primary 20D05
DOI: https://doi.org/10.1090/S0002-9947-1978-0466300-7
MathSciNet review: 0466300
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20D05

Retrieve articles in all journals with MSC: 20D05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0466300-7
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society