Conditions of Dedekindness of generalized norms in nonperiodic groups

Друшляк

2022

Carpathian Math. Publ.

The authors study relations between the properties of torsion locally nilpotent groups and their norms of Abelian non-cyclic subgroups. The impact of the norm of Abelian non-cyclic subgroups on the properties of the group under the condition of norm non-Dedekindness is investigated. It was found that for these restrictions, torsion locally nilpotent groups are finite extensions of a quasi-cyclic subgroup and the structure of such groups is completely described.

Section: Lemmamentioning

confidence: 99%

“…The authors continue to study the properties of groups depending on the properties of Σ-norm -the norm N A G of Abelian non-cyclic subgroups of a group, started in [6,[16][17][18][19][20]. The norm N A G of Abelian non-cyclic subgroups of G is a Σ-norm, provided that the system Σ…”

mentioning

confidence: 99%

Torsion locally nilpotent groups with non-Dedekind norm of Abelian non-cyclic subgroups

Друшляк

2022

Carpathian Math. Publ.

“…The intersection of normalizers of all non-cyclic Abelian subgroups of a group G (provided that the system of these subgroups is non-empty) is called the norm of non-cyclic Abelian subgroups of a group G and denoted by N A G (see e.g. [6,9]). If the norm N A G contains at least one Abelian noncyclic subgroup, then each such a subgroup is normal in N A G .…”

Section: Introductionmentioning

confidence: 99%

Locally soluble groups with the restrictions on the generalized norms

2020

ADM

“…Narrowing the system of subgroups one can get different Σ-norms which can be considered as generalizations of the norm N (G). Recently the interest to study the Σ-norms does not decrease in view of series of papers [2,3,4,7,8,9,10,11,12,13].…”

mentioning

confidence: 99%

“…The authors focused on the study of the properties of the norm N G (C p) of cyclic subgroups of non-prime order in non-periodic groups, its impact on group properties and relations with the norm N G (C ∞ ) of infinite cyclic subgroups, which is the intersection of normalizers of all infinite cyclic subgroups of a group G (see, [9,12]).…”

mentioning

confidence: 99%

Non-periodic groups with the restrictions on the norm of cyclic subgroups of non-prime order

Друшляк

2022

Mat. Stud.

One of the main directions in group theory is the study of the impact of characteristic subgroups on the structure of the whole group. Such characteristic subgroups include different $\Sigma$-norms of a group. A $\Sigma$-norm is the intersection of the normalizers of all subgroups of a system $\Sigma$. The authors study non-periodic groups with the restrictions on such a $\Sigma$-norm -- the norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order, which is the intersection of the normalizers of all cyclic subgroups of composite or infinite order of $G$. It was proved that if $G$ is a mixed non-periodic group, then its norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order is either Abelian (torsion or non-periodic) or non-periodic non-Abelian. Moreover, a non-periodic group $G$ has the non-Abelian norm $N_{G}(C_{\bar{p}})$of cyclic subgroups of non-prime order if and only if $G$ is non-Abelian and every cyclic subgroup of non-prime order of a group $G$ is normal in it, and $G=N_{G}(C_{\bar{p}})$.Additionally the relations between the norm $N_{G}(C_{\bar{p}})$ of cyclic subgroups of non-prime order and the norm $N_{G}(C_{\infty})$ of infinite cyclic subgroups, which is the intersection of the normalizers of all infinite cyclic subgroups, in non-periodic groups are studied. It was found that in a non-periodic group $G$ with the non-Abelian norm $N_{G}(C_{\infty})$ of infinite cyclic subgroups norms $N_{G}(C _{\infty})$ and $N_{G}(C _{\bar{p}})$ coincide if and only if $N_{G}(C _{\infty})$ contains all elements of composite order of a group $G$ and does not contain non-normal cyclic subgroups of order 4.In this case $N_{G}(C_{\bar {p}})=N_{G}(C_{\infty})=G$.