we characterize the rational subsets of a finite group and discuss the relations to integral cayley graphs.

‎Let $Omega$ be a class of unital‎ ‎$C^*$-algebras‎. ‎We introduce the notion of a local tracial $Omega$-algebra‎. ‎Let $A$ be an $alpha$-simple unital local tracial $Omega$-algebra‎. ‎Suppose that $alpha:Gto $Aut($A$) is an action of a finite group $G$ on $A$‎ ‎which has a certain non-simple tracial Rokhlin property‎. ‎Then the crossed product algebra‎ ‎$C^*(G,A,alpha)$ is a unital local traci...

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.

let g be a group. a subset x of g is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in x. if |x| ≥ |y | for any other set of pairwise non-commuting elements y in g, then x is said to be a maximal subset of pairwise non-commuting elements. in this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

let g be a finite group and let $gk(g)$ be the prime graph of g. we assume that $n$ is an odd number. in this paper, we show that if $gk(g)=gk(b_n(p))$, where $ngeq 9$ and $pin {3,5,7}$, then g has a unique nonabelian composition factor isomorphic to $b_n(p)$ or $c_n(p)$ . as consequences of our result, $b_n(p)$ is quasirecognizable by its spectrum and also by a new proof, the validity of a con...

we characterize strictly diagonal type of embeddings offinitary symmetric groups in terms of cardinality and the characteristic. namely, we prove thefollowing.let $kappa$ be an infinite cardinal. if$g=underset{i=1}{stackrel{infty}bigcup} g_i,$ where $ g_i=fsym(kappa n_i)$,$(h=underset{i=1}{stackrel{infty}bigcup}h_i, $ where $ h_i=alt(kappa n_i) ), $ is a group of strictly diagonal type and$xi=(...