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

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 is an element of pi(G) there exists a positive integer u(p) such that each p-element of G is of order p(i) <= p(up). A group G will be called a BSP-group if each element of G has a prime power order and for each p is an element of pi(G) there exists a positive integer v(p) such that each finite p-subgroup of G is of order p(j) <= 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 8: Let G be a BSP-group satisfying 2 is an element of pi(G). Then G is a locally finite group.