Abstract:
It is proved that, if G is a finite group with a nontrivial normal 2-subgroup Q such that G/Q≅A7 and an element of order 5 from G acts without fixed points on Q, then the extension of G by Q is splittable, Q is an elementary abelian group, and Q is the direct product of minimal normal subgroups of G each of which is isomorphic, as a G/Q-module, to one of the two 4-dimensional irreducible GF(2)A7-modules that are conjugate with respect to an outer automorphism of the group A7.
Keywords:
finite group, GF(2)A7-module, completely reducible representation, prime graph.
Citation:
A. S. Kondrat'ev, I. V. Khramtsov, “The complete reducibility of some GF(2)A7-modules”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 3, 2012, 139–143; Proc. Steklov Inst. Math. (Suppl.), 283, suppl. 1 (2013), 86–90
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\by A.~S.~Kondrat'ev, I.~V.~Khramtsov
\paper The complete reducibility of some $GF(2)A_7$-modules
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2012
\vol 18
\issue 3
\pages 139--143
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\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2013
\vol 283
\issue , suppl. 1
\pages 86--90
\crossref{https://doi.org/10.1134/S0081543813090083}
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Linking options:
https://www.mathnet.ru/eng/timm847
https://www.mathnet.ru/eng/timm/v18/i3/p139
This publication is cited in the following 5 articles:
A. S. Kondrat'ev, “Finite groups with given properties of their prime graphs”, Algebra and Logic, 55:1 (2016), 77–82
A. S. Kondratev, “O konechnykh gruppakh s nebolshim prostym spektrom, II”, Vladikavk. matem. zhurn., 17:2 (2015), 22–31
N. V. Maslova, “On the coincidence of Grünberg–Kegel graphs of a finite simple group and its proper subgroup”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 129–141
Anatoly S. Kondrat'ev, “Finite almost simple $5$-primary groups and their Gruenberg–Kegel graphs”, Sib. elektron. matem. izv., 11 (2014), 634–674
Kondratev A.S., “O konechnykh gruppakh s nebolshim prostym spektrom”, Matematicheskii forum (itogi nauki. yug Rossii), 6 (2012), 56–74