TY - JOUR

T1 - On groups in which every element has a prime power order and which satisfy some boundedness condition

AU - Herzog, Marcel

AU - Longobardi, Patrizia

AU - Maj, Mercede

N1 - Publisher Copyright:
© 2023 World Scientific Publishing Company.

PY - 2023/10/1

Y1 - 2023/10/1

N2 - 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 π(G) there exists a positive integer up such that each p-element of G is of order pi ≤ pup. A group G will be called a BSP-group if each element of G has a prime power order and for each p π(G) there exists a positive integer vp such that each finite p-subgroup of G is of order pj ≤ pvp. 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 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 π(G). Then G is a locally finite group.

AB - 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 π(G) there exists a positive integer up such that each p-element of G is of order pi ≤ pup. A group G will be called a BSP-group if each element of G has a prime power order and for each p π(G) there exists a positive integer vp such that each finite p-subgroup of G is of order pj ≤ pvp. 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 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 π(G). Then G is a locally finite group.

KW - Element orders

KW - locally finite groups

KW - locally graded groups

UR - http://www.scopus.com/inward/record.url?scp=85133880629&partnerID=8YFLogxK

U2 - 10.1142/S0219498823502171

DO - 10.1142/S0219498823502171

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85133880629

SN - 0219-4988

VL - 22

JO - Journal of Algebra and its Applications

JF - Journal of Algebra and its Applications

IS - 10

M1 - 2350217

ER -