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

CHARACTERIZATIONS OF FINITE ABELIAN AND MINIMAL NONABELIAN GROUPS

Yakov Berkovich

Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
e-mail: berkov@math.haifa.ac.il


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.


Full text (PDF) (free access)

DOI: 10.3336/gm.45.1.05


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