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In this paper as the main result, we determine finite groups with the same prime graph as the automorphism group of a sporadic simple group, except $J_2$.

There exist many results about the Diophantine equation $(q^n-1)/(q-1)=y^m$, where $m\ge 2$ and $n\ge 3$. In this paper, we suppose that $m=1$, $n$ is an odd integer and $q$ a power of a prime number. Also let $y$ be an integer such that the number of prime divisors of $y-1$ is less than or equal to $3$. Then we solve completely the Diophantine equation $(q^n-1)/(q-1)=y$ for infinitely many values of $y$. This result finds frequent applications in the theory of finite groups.

The order of every finite group $G$ can be expressed as a product of coprime positive integers $m_1,\cdots ,m_t$ such that $\pi \left(m_i\right)$ is a connected component of the prime graph of $G$. The integers $m_1,\cdots ,m_t$ are called the order components of $G$. Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups $C_2\left(q\right)$ where $q>5$ are also uniquely determined by their order components. As corollaries of this result, the validities of a...

Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $\left|G\right|$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $pq$. The prime graph of $G$ is denoted by $\Gamma \left(G\right)$. It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $\Gamma \left(G\right)=\Gamma \left(B_n\left(5\right)\right)$, where $n\ge 6$, then $G$ has a unique nonabelian composition factor isomorphic to $B_n\left(5\right)$ or $C_n\left(5\right)$.

The character degree graph of a finite group $G$ is the graph whose vertices are the prime divisors of the irreducible character degrees of $G$ and two vertices $p$ and $q$ are joined by an edge if $pq$ divides some irreducible character degree of $G$. It is proved that some simple groups are uniquely determined by their orders and their character degree graphs. But since the character degree graphs of the characteristically simple groups are complete, there are very narrow class of characteristically simple...

For a finite group $G$, the intersection graph of $G$ which is denoted by $\Gamma \left(G\right)$ is an undirected graph such that its vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H\cap K\ne 1$. In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of $\mathrm{Aut}\left(\Gamma \right(G\left)\right)$.

For a finite group $G$, $\Gamma \left(G\right)$, the intersection graph of $G$, is a simple graph whose vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H\cap K\ne 1$. In this paper, we classify all finite nonsimple groups whose intersection graphs have a leaf and also we discuss the characterizability of them using their intersection graphs.

Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $\left|\mathrm{PSL}\right(2,p^2\left)\right|$ such that $G$ has an irreducible character of degree $p^2$ and we know that $G$ has no irreducible character $\theta$ such that $2p\mid \theta \left(1\right)$, then $G$ is isomorphic to $\mathrm{PSL}(2,p^2)$. As a consequence of our result we prove that $\mathrm{PSL}(2,p^2)$ is uniquely determined by the structure of its complex group algebra.