let $g$ be a finite group‎. ‎we denote by $psi(g)$ the integer $sum_{gin g}o(g)$‎, ‎where $o(g)$ denotes the order of $g in g$‎. ‎here we show that‎ ‎$psi(a_5)< psi(g)$ for every non-simple group $g$ of order $60$‎, ‎where $a_5$ is the alternating group of degree $5$‎. ‎also we prove that $psi(psl(2,7))

let $g$ be a finite group‎. ‎we denote by $psi(g)$ the integer $sum_{gin g}o(g)$‎, ‎where $o(g)$ denotes the order of $g in g$‎. ‎here we show that‎ ‎$psi(a_5)< psi(g)$ for every non-simple group $g$ of order $60$‎, ‎where $a_5$ is the alternating group of degree $5$‎. ‎also we prove that $psi(psl(2,7))

How can we describe all finite groups? Before we address this question, let’s write down a list of all the finite groups of small orders ≤ 15, up to isomorphism. We have seen almost all of these already. If G is abelian, it is easy to write down all possible G of a given order, using the Fundamental Theorem of Finite Abelian Groups: G must be isomorphic to a direct product of cyclic groups, and...

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=(...

Abstract A graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ transitive on the set of vertices and s -arcs , where for an integer $s \ge 1$ -arc a sequence $s+1$ $(v_0,v_1,\ldots ,v_s)$ such that $v_{i-1}$ $v_i$ are adjacent $1 i s$ $v_{i-1}\ne v_{i+1}$ s-1$ . 2-transitive it $(\text{Aut} ), 2)$ but not 3)$ -arc-transitive. Cayley group G normal in $\text{Aut} n...