in this paper, we examine that some simple $k_4$-groups can be determined uniquely by their orders and one or two irreducible complex character degrees.

In this paper, it is proved that all simple $K_4$-groups of type $L_2(q)$ can be characterized by their maximum element orders together with their orders. Furthermore, the automorphism groups of simple $K_4$-groups of type $L_2(q)$ are also considered.

in this paper, it is proved that all simple $k_4$-groups of type $l_2(q)$ can be characterized by their maximum element orders together with their orders. furthermore, the automorphism groups of simple $k_4$-groups of type $l_2(q)$ are also considered.

Let $G$ be a non-abelian finite group. In this paper, we prove that $Gamma(G)$ is $K_4$-free if and only if $G cong A times P$, where $A$ is an abelian group, $P$ is a $2$-group and $G/Z(G) cong mathbb{ Z}_2 times mathbb{Z}_2$. Also, we show that $Gamma(G)$ is $K_{1,3}$-free if and only if $G cong {mathbb{S}}_3,~D_8$ or $Q_8$.

let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.

this is a survey article on centralizers of finite‎ ‎subgroups in locally finite‎, ‎simple groups or lfs-groups as we‎ ‎will call them‎. ‎we mention some of the open problems about‎ ‎centralizers of subgroups in lfs-groups and applications of the‎ ‎known information about the centralizers of subgroups to the‎ ‎structure of the locally finite group‎. ‎we also prove the‎ ‎following‎: ‎let $g$ be...

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))

Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed find asymptotic results for very large graphs, so it seems that the not suitable finding numbers. But this intuition wrong, and we will develop technique do just in paper. We new upper bounds many graph hypergraph As result, prove values $R(K_4^-,K_4^-,K_4^-)=28$, $R(...