Abstract:

Let $G$ be a finite group, $\pi(G)$ is a set of prime divisors of its orders, $\omega(G)$ is a set of orders of its elements
(its spectra).

The prime graph (or the Gruenberg–Kegel graph) of a finite group $G$ is a simple graph $GK(G)$ whose vertices
are the prime divisors of the order of $G$, and two distinct vertices $p$ and $q$ are adjacent in $GK(G)$ if and only if $G$
contains an element of order $pq$.  The prime graphs of non-abelian finite simple groups are known.
One of the most popular fields of research in finite group theory is study of finite groups by 
their prime graphs. 
We study composition factors of finite groups whose the prime graphs are the same as the prime graphs of  non-abelian finite simple groups. In the paper, we consider the question about existence of sporadic composition factors in such finite groups. As finite simple groups we take some series of exceptional and classical groups of Lie type, and also alternating groups.

In 2011, A. M. Staroletov studied finite groups with spectrum like a finite simple non-abelian group that have a sporadic composition factor. Generalizing this result, we consider the question of whether the composition factor of a finite group with the prime graph like a finite simple non-abelian group can be isomorphic to a sporadic group. It is shown that a finite group with the prime graph 
like the classical groups $L_n(q)$, $U_n(q)$, where $n\geq 11$, $H\in\{O^{+}_{2n}(q),O^{-}_{2n}(q)\}$, 
where $n\geq 9$, has no sporadic composition factors. For finite groups with the prime graph like the exceptional groups ${^3}D_4(q)$ и $G_2(q)$ we find small list  of possible sporadic composition factors of such groups.