AbstractLet W be a non-empty subset of a free group. The automorphism of a group G is said to be a marginal automorphism, if for all x in G,x^−1alpha(x) in W^*(G), where W^*(G) is the marginal subgroup of G.In this paper, we give necessary and sufficient condition for a purelynon-abelian p-group G, such that the set of all marginal automorphismsof G forms an elementary abelian p-group.

abstractlet w be a non-empty subset of a free group. the automorphism of a group g is said to be a marginal automorphism, if for all x in g,x^−1alpha(x) in w^*(g), where w^*(g) is the marginal subgroup of g.in this paper, we give necessary and sufficient condition for a purelynon-abelian p-group g, such that the set of all marginal automorphismsof g forms an elementary abelian p-group.

Let W be a variety of groups defined by a set W of laws and G be a finite p-group in W. The automorphism α of a group G is said to bea marginal automorphism (with respect to W), if for all x ∈ G, x−1α(x) ∈ W∗(G), where W∗(G) is the marginal subgroup of G. Let M,N be two normalsubgroups of G. By AutM(G), we mean the subgroup of Aut(G) consistingof all automorphisms which centralize G/M. AutN(G) ...

In this paper we look at the K-theory of a specific C*-algebra closely related to the irrational rotation algebra. Also it is shown that any automorphism of a C*-algebra A induces group automorphisms of K_{1}(A) amd K_{0}(A) in an obvious way. An interesting problem for any C*-algebra A is to find out whether, given an automorphism of K_{0}(A) and an automorphism of K_{1}(A), we can lift them t...

The conformal algebra in 2D (Diff($S^{1}$)$\oplus$Diff($S^{1}$)) is shown to be preserved under a nonlinear map that mixes both chiral (holomorphic) generators $T$ and $\bar{T}$. It depends on single real parameter it can regarded as ``nonlinear $SO(1,1)$ automorphism.'' preserves the form of momentum density naturally induces flow energy by marginal $\sqrt{T\bar{T}}$ deformation. In turn, gene...

This paper aims to explore the inherent connection between Heisenberg groups, quantum Fourier transform (QFT) and (quasi-probability) distribution functions. Distribution functions for continuous and finite quantum systems are examined from three perspectives and all of them lead to Weyl–Gabor–Heisenberg groups. The QFT appears as the intertwining operator of two equivalent representations aris...

a longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$‎. ‎let $g$ be a finite nonabelian $p$-group‎. ‎it is known that if $g$ is regular or of nilpotency class $2$ or the commutator subgroup of $g$ is cyclic‎, ‎or $g/z(g)$ is powerful‎, ‎then $g$ has a noninner automorphism of order $p$ leaving either the center $z(g)$ or the frattin...

a longstanding conjecture asserts that every finite nonabelian $p$-group admits a noninner automorphism of order $p$. let $g$ be a finite nonabelian $p$-group. it is known that if $g$ is regular or of nilpotency class $2$ or the commutator subgroup of $g$ is cyclic, or $g/z(g)$ is powerful, then $g$ has a noninner automorphism of order $p$ leaving either the center $z(g)$ or the frattini subgro...

in this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. with this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. n(g) and s(g) are the set of all nilpotency classes and the set of all solvability classes for the group g with respect to different automorphi...

let be a set and let be the set of subsets of . the pair in which is a collection of elements of (blocks) is called a design if every element of appears in , times. is called a symmetric design if . in a symmetric design, each element of appears times in blocks of . a mapping between two designs and is an isomorphism if is a one-to-one correspondence and . every isomorphism of a design, , to it...