A finiteness condition on centralizers in locally finite groups

2015/10/13 by Fernandez-Alcober, Gustavo A., Legarreta, Leire, Tortora, Antonio +1
#20E34 #20F50 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1510.03733

Abstract

We consider a finiteness condition on centralizers in a group G, namely that |C_G (x) : | is finite for every non-normal cyclic subgroup of G. For periodic groups, this is the same as |C_G (x)| is finite for every non-normal cyclic subgroup of G. We give a full description of locally finite groups satisfying this condition. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study a variation of our condition, where the requirement of finiteness is replaced with a bound: |C_G (x) : | < n for every non-normal cyclic subgroup of G, for some fixed n. In this case, we are able to extend our analysis to the class of periodic locally graded groups.

Related

• Some finiteness conditions on normalizers or centralizers in groups
• Locally finite groups in which every non-cyclic subgroup is self-centralizing
• On a problem of B. Hartley about a small centralizer in finite and locally finite groups
• Locally finite groups of finite centraliser dimension
• On n-centralizer CA-groups