Groups with a Finite Number of Normalizer Subgroups

On Solubility of Groups With Few Normalisers

“…All finite groups having exactly two normalisers were classified by Pérez-Ramos [5]; Camp-Mora [1] generalised that result to locally finite groups. In 2004, Tota [6] showed that every group with at most four normalisers of subgroups is soluble of derived length at most two.…”

On Groups With a Finite Number of Normalisers

“…Pérez-Ramos [13] characterised finite groups belonging to N 2 , and then Camp-Mora [5] generalised this result to locally finite groups. Subsequently Tota [15] investigated the behaviour of normaliser subgroups of a group on the structure of the group itself and gave some properties of arbitrary groups with two, three and four normalisers. More precisely, among other things, her results can be described in the following way.…”

“…This result suggests that the behaviour of normalisers has a strong influence on the structure of the group. In fact Tota [15] (see Theorem 2.1 and also the remark after Theorem 2.2) has proved that a group G has finitely many normalisers of cyclic subgroups if and only if G is a centralby-finite group if and only if G has finitely many normalisers of subgroups. Therefore we can summarise the latter results in the following theorem.…”

On solubility of groups with finitely many centralizers

“…Also they characterized all finite C n -groups for n ∈ {4, 5}. Tota (see Appendix of [10]) proved that every arbitrary C 4 -group is soluble. The author in [11] showed that the derived length of a soluble C n -group (not necessarily finite) is ≤ n.…”