Finite groups with the same conjugacy class sizes as a finite simple group

For a finite group $H$‎, ‎let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$‎. ‎In this paper‎, ‎we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$‎, ‎then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$‎. ‎In particular‎, ‎we show that for some finite simple group $S$‎, ‎$G cong S times Z(G)$‎

pp-parts of co-degrees of irreducible characters

For a character $\chi$ of a finite group $G$, the co-degree of $\chi$ is $\chi ^c(1)=\frac{[G:\ker \chi ]}{\chi (1)}$. Let $p$ be a prime and let $e$ be a positive integer. In this paper, we first show that if $G$ is a $p$-solvable group such that $p^{e+1}\nmid \chi ^c(1)$, for every irreducible character $\chi$ of $G$, then the $p$-length of $G$ is not greater than $e$. Next, we study the finite groups satisfying the condition that $p^2$ does not divide the co-degrees of their irreducible characters

ON THE GROUPS WITH THE PARTICULAR NON-COMMUTING GRAPHS

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 $Q_8$

pp-parts of co-degrees of irreducible characters

For a character $\chi$ of a finite group $G$, the co-degree of $\chi$ is $\chi ^c(1)=\frac{[G:\ker \chi ]}{\chi (1)}$. Let $p$ be a prime and let $e$ be a positive integer. In this paper, we first show that if $G$ is a $p$-solvable group such that $p^{e+1}\nmid \chi ^c(1)$, for every irreducible character $\chi$ of $G$, then the $p$-length of $G$ is not greater than $e$. Next, we study the finite groups satisfying the condition that $p^2$ does not divide the co-degrees of their irreducible characters

Quasirecognition by prime graph of Cn(4)C_{n}(4) , where n≥17 n\geq17 is odd

Let $G$ be a finite group and let $\Gamma(G)$ be the prime graph of $G$. We assume that $n\geq 17$ is an odd number. In this paper, we show that if $\Gamma(G) = \Gamma(C_{n}(4))$, then $G$ has a unique non-abelian composition factor isomorphic to $C_{n}(4)$. As consequences of our result, $C_{n}(4)$ is quasirecognizable by its spectrum and by prime graph

Characterization of some simple K4K_4-groups by some irreducible complex character degrees

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