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

Marcel Herzog; Patrizia Longobardi; Mercede Maj

paper

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

arXiv:2205.07248

Abstract

In this paper we shall deal with periodic groups, in which each element has a prime power order. A group $G$ will be called a $B C P$ -group if each element of $G$ has a prime power order and for each $p \in π (G)$ there exists a positive integer $u_{p}$ such that each $p$ -element of $G$ is of order $p^{i} \leq p^{u_{p}}$ . A group $G$ will be called a $B S P$ -group if each element of $G$ has a prime power order and for each $p \in π (G)$ there exists a positive integer $v_{p}$ such that each finite $p$ -subgroup of $G$ is of order $p^{j} \leq p^{v_{p}}$ . Here $π (G)$ denotes the set of all primes dividing the order of some element of $G$ . Our main results are the following four theorems. Theorem 1: Let $G$ be a finitely generated $B C P$ -group. Then $G$ has only a finite number of normal subgroups of finite index. Theorem 4: Let $G$ be a locally graded $B C P$ -group. Then $G$ is a locally finite group. Theorem 7: Let $G$ be a locally graded $B S P$ -group. Then $G$ is a finite group. Theorem 9: Let $G$ be a $B S P$ -group satisfying $2 \in π (G)$ . Then $G$ is a locally finite group.