On groups in which every element has a prime power order and which satisfy some boundedness condition

Abstract

In this paper, we shall deal with periodic groups, in which each element has a prime power order. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each [Formula: see text]-element of [Formula: see text] is of order [Formula: see text]. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each finite [Formula: see text]-subgroup of [Formula: see text] is of order [Formula: see text]. Here, [Formula: see text] denotes the set of all primes dividing the order of some element of [Formula: see text]. Our main results are the following four theorems. Theorem 1: Let [Formula: see text] be a finitely generated [Formula: see text]-group. Then [Formula: see text] has only a finite number of normal subgroups of finite index. Theorem 4: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a locally finite group. Theorem 7: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a finite group. Theorem 8: Let [Formula: see text] be a [Formula: see text]-group satisfying [Formula: see text]. Then [Formula: see text] is a locally finite group.