Solvability of finite groups

Abstract

H is called an {{\displaystyle \mathcal{M}}}_p-embedded subgroup of G, if there exists a pnilpotent subgroup B of G such that H_p ∈ Syl_p (B) and B is {{\displaystyle \mathcal{M}}}_p-supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use {{\displaystyle \mathcal{M}}}_p-embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that \mathrm{1}\langle d\le \left|P\right| and d divides |P|. If every subgroup H of P with \left|H\right|=d is {{\displaystyle \mathcal{M}}}_5-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5'-group, (3) I/C\cong A_5

Key words： Composition factor; {{\displaystyle \mathcal{M}}}_p-embedded subgroup; primary subgroup; solvable

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