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Let M be a subgroup of a finite group G. We define the core of M in G by Cor G (M) = ∩ g∈G M g . Clearly, Cor G (M) is the largest normal subgroup of G contained in M. We determine the structure of a finite group G if G possesses a maximal subgroup with core 1 and all maximal subgroups M of G with Cor G (M) = 1 satisfy a certain property.
Communicated by M. R. Dixon.
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