let $gamma$ be a normal subgroup of the full automorphism group $aut(g)$ of a group $g$‎, ‎and assume that $inn(g)leq gamma$‎. ‎an endomorphism $sigma$ of $g$ is said to be {it $gamma$-central} if $sigma$ induces the the identity on the factor group $g/c_g(gamma)$‎. ‎clearly‎, ‎if $gamma=inn(g)$‎, ‎then a $gamma$-central endomorphism is a {it central} endomorphism‎. ‎in this article the conditi...

We prove that the automorphism group of any non-abelian free group F is complete. The key technical step in the proof: the set of all conjugations by powers of primitive elements is first-order parameter-free definable in the group Aut(F ). Introduction In 1975 J. Dyer and E. Formanek [2] had proved that the automorphism group of a finitely generated non-abelian free group F is complete (that i...

An almost Abelian Lie group is a non-Abelian with codimension 1 normal subgroup. The majority of 3-dimensional real groups are Abelian, and they appear in all parts physics that deal anisotropic media—cosmology, crystallography etc. In theoretical differential geometry, their homogeneous spaces provide some the simplest solvmanifolds on which variety geometric structures, such as symplectic, Kä...

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.

Let r be a finite connecied regular bipartite 2-arc transitive graph. It is shown that r is a cover of a possibly smaller graph :E, which is also connecied and regular of the same valency as r, and there is a subgroup G of Aut :E such that G is 2-arc transitive on :E and every nontrivial normal subgroup of G has at most two orbits on vertices. Such graphs :E for which the subgroup G has an abel...

1. The dihedral subgroup is non-abelian. If a group G has the property that the squares of its operators constitute a given group then the direct product of G and any abelian group of order 2m and of type (1,1, 1, • • • ) has the same property. Such direct products will always be excluded in what follows. It has been noted that a necessary and sufficient condition that there is at least one gro...

Let G be a group. We denote by ν(G) certain extension of the non-abelian tensor square G⊗G G×G. In this paper we prove that derived subgroup ν(G)′ is central product three normal subgroups ν(G), all isomorphic to G⊗G. As consequence, describe structure each term and lower series group ν(G).

A graph is edge-primitive if its automorphism group acts primitively on the edge set of graph. Edge-primitive graphs form an important subclass symmetric graphs. In this paper, order as a product two distinct primes are completely determined. This depends non-abelian simple groups with subgroup index pq being classified, where p>q odd primes.

Let be a non-discrete LCA (locally compact abelian) group, its dual. ( ) ∧, where denotes discrete, is the Bohr compactification of , denoted by . There exists a continuous group isomorphism ί: → of onto a proper dense subgroup of . This subgroup to be denoted by ί is the main object of our study. A subgroup of is pure if ∩ = for each positive integer . We define to be strongly pure if is pure ...

A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {x ∈ X : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class o...