Some finiteness conditions on normalizers or centralizers in groups

Gustavo A. Fernandez-Alcober, Leire Legarreta, Antonio Tortora, Maria Tota

Published 2014-12-23Version 1

We consider several finiteness conditions on normalizers and centralizers in a group G, the most important of which are the following two: (i) the index |N_G(H):H| is finite for every non-normal subgroup H of G, and (ii) the index |C_G(x):<x>| is finite for every non-normal cyclic subgroup <x> of G. We show that (i) and (ii) are equivalent in the realms of locally finite groups and locally nilpotent groups, in which case we can give a full description of the groups satisfying these conditions. As it turns out, they are a special type of cyclic extensions of Dedekind groups. We also study the following variations of conditions (i) and (ii), where the requirement of finiteness is replaced with a bound: (i') |N_G(H):H| <= n for every non-normal subgroup H of G, and (ii') |C_G(x):<x>| <= n for every non-normal cyclic subgroup <x> of G, for some fixed n. These conditions are equivalent to (i) and (ii) for locally finite groups, but on the contrary, they imply that the group G is abelian for non-periodic locally nilpotent groups. Also, we are able to extend our analysis of conditions (i') and (ii') to the classes of non-periodic groups and periodic locally graded groups.