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 pn≠2. 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.
