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.

Some finiteness conditions on normalizers or centralizers in groups

Antonio Tortora;Maria Tota
2018-01-01

Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4705021
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