Let $G$ be a finite group and $pi(G)$ be the set of all prime divisors of $|G|$. The prime graph of $G$ is a simple graph $Gamma(G)$ with vertex set $pi(G)$ and two distinct vertices $p$ and $q$ in $pi(G)$ are adjacent by an edge if an only if $G$ has an element of order $pq$. In this case, we write $psim q$. Let $|G= p_1^{alpha_1}cdot p_2^{alpha_2}cdots p_k^{alpha_k}$, where $p_1

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.

Vasil'ev posed Problem 16.26 in [The Kourovka Notebook: Unsolved Problems in Group Theory, 16th ed.,Sobolev Inst. Math., Novosibirsk (2006).] as follows:Does there exist a positive integer $k$ such that there are no $k$ pairwise nonisomorphicnonabelian finite simple groups with the same graphs of primes? Conjecture: $k = 5$.In [Zvezdina, On nonabelian simple groups having the same prime graph a...

 After the classification of the flag-transitive linear spaces, the attention has been turned to line-transitive linear spaces. In this article, we present a partial classification of the finite linear spaces $mathcal S$ on which an almost simple group $G$ with the socle $G_2(q)$ acts line-transitively.