Large subgroups of a finite group of even order

Amberg, Bernhard; Kazarin, Lev

doi:10.1090/S0002-9939-2011-10982-1

Large subgroups of a finite group of even order
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by Bernhard Amberg and Lev Kazarin PDF

Proc. Amer. Math. Soc. 140 (2012), 65-68 Request permission

Abstract:

It is shown that if $G$ is a group of even order with trivial center such that $|G|>2|C_{G}(t)|^{3}$ for some involution $t\in G$, then there exists a proper subgroup $H$ of $G$ such that $|G|< |H|^{2}$. If $|G|>|C_{G}(t)|^{3}$ and $k(G)$ is the class number of $G$, then $|G|\leq k(G)^{3}$.

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