Locally finite groups with two normalizers

On Groups With a Finite Number of Normalisers

“…The groups belonging to N 1 are the Dedekind groups, which are well known. 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.…”

“…If G is isomorphic to L 2 (3 m ) (L 2 (5 m ), respectively), m a prime, then again, by Proposition 4.2, we can see that n ≥ 92 (n ≥ 652, respectively), a contradiction. Thus among the projective special linear groups, we only need to investigate the groups L 3 (3) and L 3 (5). If G L 3 (3), then |G| = 2 4 × 3 3 × 13 and v 13 (G) = 144.…”

“…So n ≥ 145, by Lemma 4.5, a contradiction. If G L 3 (5), then |G| = 5 3 × 2 5 × 3 × 31 and v 31 (G) = 4000. Thus n ≥ 4001, by Lemma 4.5, a contradiction.…”

On Solubility of Groups With Few Normalisers

“…The groups having exactly one normaliser are the Dedekind groups. 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.…”