Summary
Dπ-property (π=set of primes) in finite groups is not in general inherited by subgroups. In this paper, as evidence in favor of the following conjecture (F. Gross): (o) If a finite group G satisfies Dπ then its normal subgroups satisfy Dπ-property as well. the Author shows that if the Dπ and the Dπ-properties (π′=set of the primes not in π) hold together in a finite group G, then both are inherited by the normal subgroups of G. As a corollary, the characterization of the groups satisfying both the properties Dπ and Dπ′ is given in terms of the composition factors.
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Gilotti, A.L. 227-01227-01227-01-property and normal subgroups. Annali di Matematica pura ed applicata 148, 227–235 (1987). https://doi.org/10.1007/BF01774290
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DOI: https://doi.org/10.1007/BF01774290