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 study the longstanding conjecture of whether there exists a noninner automorphism of order p for a finite non-abelian pgroup. Among other results, we prove that if G is a finite non-abelian pgroup, p is odd and G/Z(G) is powerful then G has a noninner automorphism of order p. To prove the latter result we show that the Tate cohomology Hn(G/N, Z(N)) 6= 0 for all n ≥ 0, where G i...

We prove that for any prime number p, every finite non-abelian p-group G of class 2 has a noninner automorphism of order p leaving either the Frattini subgroup Φ(G) or Ω1(Z(G)) elementwise fixed.

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...

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...

In this paper, we consider the projective special linear group $PSL_2(59)$ and construct some 1-designs by applying the Key-Moori method on $PSL_2(59)$. Moreover, we obtain parameters of these designs and their automorphism groups. It is shown that $PSL_2(59)$ and $PSL_2(59):2$ appear as the automorphism group of the constructed designs.