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

TL;DR

This paper investigates groups where every element has a prime power order and satisfy boundedness conditions, establishing finiteness and local finiteness properties for various classes of such groups.

Contribution

It introduces and analyzes the classes of BCP and BSP groups, proving finiteness and local finiteness results under specific conditions, extending understanding of their structure.

Findings

Finitely generated BCP-groups have finitely many normal subgroups of finite index.

Locally graded BCP-groups are locally finite.

BSP-groups with certain conditions are finite or locally finite.

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 $BCP$-group if each element of $G$ has a prime power order and for each $p\in \pi(G)$ 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 $BSP$-group if each element of $G$ has a prime power order and for each $p\in \pi(G)$ 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 $\pi(G)$ 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 $BCP$-group. Then $G$ has only a finite number of normal subgroups of finite index. Theorem 4: Let $G$ be a locally graded $BCP$-group. Then $G$ is a locally finite group. Theorem 7: Let $G$ be a locally graded $BSP$-group. Then $G$ is a finite group. Theorem 9: Let $G$ be a $BSP$-group satisfying $2\in \pi(G)$. Then $G$ is a locally finite group.