Groups with complemented infinite Abelian subgroups

Abstract

The author has shown that in the class of binary-finite groups this question has an affirmative answer, i.e., any binary-finite group with complemented infinite Abelian subgroups is locally finite (see [5, 6]). However, as Theorem 1 of the present paper shows, in general the answer to this question is negative. Moreover, it follows from this theorem that a periodic group containing infinite Abelian subgroups need not be locally finite under the assumption that all noncyclic Abelian subgroups or even all Abelian subgroups of nonprime orders are complemented. Note that the locally finite groups in which Abelian noncyclic subgroups are complemented and the locally finite groups with complemented Abelian subgroups of nonprime orders were described in [7] and [8], and the locally finite groups with complemented elementary Abelian subgroups of nonprime orders in [9] and [i0].