Abstract
H is called an ℳ p -embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H p ∈ Syl p (B) and B is ℳ p -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use 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 1 < d ⩽ |P| and d divides |P|. If every subgroup H of P with |H| = d is ℳ5-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 ≅ A5.
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