Glasnik Matematicki, Vol. 45, No.1 (2010), 55-62.

CHARACTERIZATIONS OF FINITE ABELIAN AND MINIMAL NONABELIAN GROUPS

Yakov Berkovich

Abstract.   In this note we present the following characterizations of finite abelian and minimal nonabelian groups:
(i) A group G is abelian if and only if G' = Φ(G)'.
(ii) A group G is either abelian or minimal nonabelian if and only if Φ(G)' = H' for all maximal subgroups H of G.
We also prove a number of related results.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Maximal subgroup, abelian, minimal nonabelian, minimal nonnilpotent and Frobenius groups, Frattini subgroup, derived subgroup.

DOI: 10.3336/gm.45.1.05

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