Groups with finitely many normalizers of non-subnormal subgroups (original) (raw)

Groups with finitely many normalizers of subnormal subgroups

Journal of Algebra, 2006

The structure of soluble groups in which normality is a transitive relation is known. Here, groups with finitely many normalizers of subnormal subgroups are investigated, and the behavior of the Wielandt subgroup of such groups is described; moreover, groups having only finitely many normalizers of infinite subnormal subgroups are considered.

Groups with Few Normalizer Subgroups

The behaviour of normalizer subgroups of a group has often a strong influence on the structure of the group itself. In this paper groups with finitely many normalizers of subgroups with a given property χ are investigated, for various relevant choices of the property χ.

On Groups with a Finite Number of Normalisers

Bulletin of the Australian Mathematical Society, 2012

Groups having exactly one normaliser are well known. They are the Dedekind groups. All finite groups having exactly two normalisers were classified by Pérez-Ramos [‘Groups with two normalizers’, Arch. Math.50 (1988), 199–203], and Camp-Mora [‘Locally finite groups with two normalizers’, Comm. Algebra28 (2000), 5475–5480] generalised that result to locally finite groups. Then Tota [‘Groups with a finite number of normalizer subgroups’, Comm. Algebra32 (2004), 4667–4674] investigated properties (such as solubility) of arbitrary groups with two, three and four normalisers. In this paper we prove that every finite group with at most 20 normalisers is soluble. Also we characterise all nonabelian simple (not necessarily finite) groups with at most 57 normalisers.

Groups with Finitely Many Normalizers of Non-Nilpotent Subgroups

Mathematical Proceedings of the Royal Irish Academy, 2007

It is known that (generalized) soluble groups in which every non-normal subgroup is locally nilpotent either are locally nilpotent or have a finite commutator subgroup. Here the structure of (generalized) soluble groups with finitely many normalizers of (infinite) non-(locally nilpotent) subgroups is investigated, and the above result is extended to this more general situation.

On groups with a restriction on normal subgroups

International Journal of Group Theory, 2018

The structure of infinite groups in which every (proper) normal subgroup is the only one of its cardinality is investigated in the universe of groups without infinite simple sections. The corrisponding problem for finite soluble groups was considered by M. .

Structure of Groups with Generalized Normal Subgroups

The present paper deals with a subgroup X of a group G is almost normal if the index |G: NG(X)| is finite, while X is nearly normal if it has finite index in the normal closure XG. This paper investigates the structure of groups in which every (infinite) subgroup is either almost normal or nearly normal.