s^∗-normality and coprime action in finite groups

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Abstract

A subgroup H of a finite group G is called s^∗-normal in G if some normal subgroup T of G satisfies H_G ≤ T, G = HT, and the quotient (H ∩ T)/H_G has square-free order, where H_G is the core of H in G. This paper gave several new criteria for finite groups to be solvable by considering the s^∗-normality of certain n-maximal invariant subgroups under coprime actions, where n ∈{1, 2, 3}.

Keywords

solvable group / n-maximal subgroup / s^∗-normal subgroup / coprime action

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