Vol. 31, No. 1, 1969

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Centralizers of abelian, normal subgroups of hypercyclic groups

Ulrich F. K. Schoenwaelder

Vol. 31 (1969), No. 1, 197–208

J. L. Alperin proved the following theorem about finite p-groups G: if E is maximal among the abelian, normal subgroups of G of exponent dividing pn, then ΩnC𝜃(E) = E, provided that pn2. It turns out that the restriction to p-groups and also to finite groups in Alperin’s proof is not essential. In fact a similar theorem holds in a large class of hypercyclic groups (Theorem 2.2). By the same method also a modified version (Theorem 2.8) will be obtained, the word “normal” in the assumptions about E being replaced by “characteristic”, here G is supposed to be hypercentral; the modification results in enlarging E to a characteristic subgroup k𝜃(E) of class 2 in a very definite way before taking its centralizer.

Mathematical Subject Classification
Primary: 20.40
Received: 18 November 1968
Published: 1 October 1969
Ulrich F. K. Schoenwaelder