on $m^{th}$-autocommutator subgroup of finite abelian groups
Authors
abstract
let $g$ be a group and $aut(g)$ be the group of automorphisms of$g$. for any naturalnumber $m$, the $m^{th}$-autocommutator subgroup of $g$ is definedas: $$k_{m}(g)=langle[g,alpha_{1},ldots,alpha_{m}] |gin g,alpha_{1},ldots,alpha_{m}in aut(g)rangle.$$in this paper, we obtain the $m^{th}$-autocommutator subgroup ofall finite abelian groups.
similar resources
On $m^{th}$-autocommutator subgroup of finite abelian groups
Let $G$ be a group and $Aut(G)$ be the group of automorphisms of $G$. For any natural number $m$, the $m^{th}$-autocommutator subgroup of $G$ is defined as: $$K_{m} (G)=langle[g,alpha_{1},ldots,alpha_{m}] |gin G,alpha_{1},ldots,alpha_{m}in Aut(G)rangle.$$ In this paper, we obtain the $m^{th}$-autocommutator subgroup of all finite abelian groups.
full textsubgroup intersection graph of finite abelian groups
let $g$ be a finite group with the identity $e$. the subgroup intersection graph $gamma_{si}(g)$ of $g$ is the graph with vertex set $v(gamma_{si}(g)) = g-e$ and two distinct vertices $x$ and $y$ are adjacent in $gamma_{si}(g)$ if and only if $|leftlangle xrightrangle capleftlangle yrightrangle|>1$, where $leftlangle xrightrangle $ is the cyclic subgroup of $g$ generated by $xin g$. in th...
full textOn non-normal non-abelian subgroups of finite groups
In this paper we prove that a finite group $G$ having at most three conjugacy classes of non-normal non-abelian proper subgroups is always solvable except for $Gcong{rm{A_5}}$, which extends Theorem 3.3 in [Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable, Acta Math. Sinica (English Series) 27 (2011) 891--896.]. Moreover, we s...
full textA characterization of subgroup lattices of finite Abelian groups
We show that every primary lattice can be considered a glueing of intervals having geometric dimension at least 3 and with a skeleton of breadth at most 2. We call this geometric decomposition. In the Arguesian case, we analyse the sub-glueings corresponding to cover preserving sublattices of the skeleton which are 2-element chains or a direct product of 2 such. We show that these admit a cover...
full textRelative non-Normal Graphs of a Subgroup of Finite Groups
Let G be a finite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or...
full textFinite $p$-groups and centralizers of non-cyclic abelian subgroups
A $p$-group $G$ is called a $mathcal{CAC}$-$p$-group if $C_G(H)/H$ is cyclic for every non-cyclic abelian subgroup $H$ in $G$ with $Hnleq Z(G)$. In this paper, we give a complete classification of finite $mathcal{CAC}$-$p$-groups.
full textMy Resources
Save resource for easier access later
Journal title:
journal of linear and topological algebra (jlta)جلد ۵، شماره ۰۲، صفحات ۱۳۵-۱۴۴
Hosted on Doprax cloud platform doprax.com
copyright © 2015-2023