Some finiteness conditions on normalizers or centralizers in groups

We consider the following two finiteness conditions on normalizers and centralizers in a group G: (i) |N_G(H) : H| < ∞ for every H ⋪ G, and (ii) |C_G(x):⟨x⟩|<∞ for every ⟨x⟩⋪G. We show that (i) and (ii) are equivalent in the classes of locally finite groups and locally nilpotent groups. In both cases, the groups satisfying these conditions are a special kind of cyclic extensions of Dedekind groups. We also study a variation of (i) and (ii), where the requirement of finiteness is replaced with a bound. In this setting, we extend our analysis to the classes of periodic locally graded groups and non-periodic groups. While the two conditions are still equivalent in the former case, in the latter the condition about normalizers is stronger than that about centralizers.