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Indices of Maximal Invariant Subgroups and Solvability of Finite Groups
(Springer Verlag, 2019)
Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We study the solvability and supersolvability of G when certain proper maximal A-invariant subgroups of
G have prime index or when ...
Simplicity of normal subgroups and conjugacy class sizes
(Springer, 2013)
Given a finite group G which possesses a non-abelian simple normal subgroup N having exactly four G -class sizes, we prove that N is isomorphic to PSL (2,2a) with a≥2 . Thus, we obtain an extension for normal subgroups of ...
Class sizes of prime-power orden p'-elements and normal subgroups
(Springer, 2014-06)
We prove an extension of the renowned Itô’s theorem on groups having two class sizes in three different directions at the same time: normal subgroups, p' -elements and prime- power order elements. Let N be a normal subgroup ...
Conditions for Sylow 2-subgroups of the Fixed Point Subgroup Implying Solubility
(Cambridge University Press, 2018)
Let A and G be finite groups and suppose that A acts via automorphisms
on G with (|A|, |G|) = 1. We study how certain conditions on the
Sylow 2-subgroups of the fixed point subgroup of the action, CG(A), may
imply the ...
Arithmetical Conditions on Invariant Sylow Numbers
(Springer Verlag, 2018)
Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. A natural number n is said to be an A-invariant Sylow number if n is the number of A-invariant Sylow p-subgroups of G for some prime ...
Landau's theorem on conjugacy classes for normal subgroups
(World Scientific, 2016-10-10)
Landau’s theorem on conjugacy classes asserts that there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes for any positive integer k. We show that, for any positive integers n and ...
Triangles in the graph of conjugacy classes of normal subgroups
(Springer, 2015-12)
Let G be a finite group and N a normal subgroup of G. We determine the structure of N when the graph ΓG(N), which is the graph associated to the conjugacy classes of G contained in N, has no triangles and when the graph ...
The structure of finite groups with three class sizes
(Walter de Gruyter, 2009)
Let G be a finite group and suppose that the set of conjugacy class sizes of G is f1; m; mng, where m; n > 1 are coprime. We prove that m 1⁄4 p for some prime p dividing n 1. We also show that G has an abelian normal ...
Nilpotency of normal subgroups having two G-class sizes
(American Mathematical Society, 2010-12-22)
Let G be a finite group. If N is a normal subgroup which has exactly two G-conjugacy class sizes, then N is nilpotent. In particular, we show that N is abelian or is the product of a p-group P by a central subgroup of G. ...
On the Solvability of groups with four class sizes
(World Scientific Publishing, 2012)
It is shown that if the set of conjugacy class sizes of a nite group G is
f1; m; n;mng, where m; n are positive integers which do not divide each
other, then G is up to central factors a fp; qg-group. In particular, G ...