let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

In this paper, we show that good structured codes over non-Abelian groups do exist. Specifically, we construct codes over the smallest non-Abelian group D6 and show that the performance of these codes is superior to the performance of Abelian group codes of the same alphabet size. This promises the possibility of using non-Abelian codes for multi-terminal settings where the structure of the cod...

Let G be a group and CayP (G) < Sym(G) be the subgroup of all permutations that induce graph automorphisms on every Cayley graph of G. The group G is graphically abelian if the map ν : g → g−1 belongs to CayP (G); these groups have been classified. Also G is irregular if there exists σ ∈ CayP (G) such that σ = 1G, σ(1) = 1 and σ = ν. We show G is irregular if and only if G = Dic(A, I); every no...

In this paper, we show how certain metabelian groups can be found within polynomial evaluation groupoids. We show that every finite abelian group can beobtained as a polynomial evaluation groupoid.

the triple factorization of a group $g$ has been studied recently showing that $g=aba$ for some proper subgroups $a$ and $b$ of $g$, the definition of rank-two geometry and rank-two coset geometry which is closely related to the triple factorization was defined and calculated for abelian groups. in this paper we study two infinite classes of non-abelian finite groups $d_{2n}$ and $psl(2,2^{n})$...

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the e...

A well known theorem of G. A. Miller [4] (see also [2]) shows that a p-group of order p" where n > v(v 1)/2 contains an Abelian subgroup of order p° . It is clear that this theorem together with Sylow's Theorem implies that any finite group of large order contains an Abelian p-group of large order . In this note we use simple number theoretic considerations to make this implication more precise...

This paper provides a thorough explication of the Structure Theorem for Abelian groups and of the background information necessary to prove it. The outline of this paper is as follows. We first consider some theorems related to abelian groups and to R-modules. In this section we see that every finitely generated abelian group is the epimorphic image of a finitely generated free abelian group. H...

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.

A group G is said to be n-centralizer if its number of element centralizers $$\mid {{\,\mathrm{Cent}\,}}(G)\mid =n$$ , an F-group every non-central centralizer contains no other and a CA-group all are abelian. For any non-abelian G, we prove that \frac{G}{Z(G)}\mid \le (n-2)^2$$ $$n 12$$ 2(n-4)^{{log}_2^{(n-4)}}$$ otherwise, which improves earlier result. We arbitrary F-group, then gcd $$(n-2, ...