On finite groups with small simple spectrum, II

This is a survey of the results about finite groups whose prime graphs have a small number of vertices obtained recently by the author jointly with his pupils. It is refined a description of the chief factors of $4$-primary groups, whose prime graphs are disconnected. The finite almost simple $5$-primary and $6$-primary groups and their Gruenberg–Kegel graphs are determined. The chief factors of the commutator subgroups of finite non-solvable groups $G$ with disconnected Gruenberg–Kegel graph having exactly $5$ vertices are described in the case when $G/F(G)$ is an almost simple $n$-primary group for $n\le4$. The problem of the realizability of a graph with at most five vertices as the prime graph of a finite group is solved. The finite almost simple groups with prime graphs, whose the connected components are complete graphs, are determined. The finite almost simple groups whose prime graphs do not contain triangles are determined. It is proved that the groups $^2E_6(2)$, $E_7(2)$ and $E_7(3)$ are recognizable by the prime graph. Absolutely irreducible $SL_n(p^f)$-modules over a field of prime characteristic $p$, where an element of a given prime order $m$ from a Zinger cycle of $SL_n(p^f)$ acts freely, are classified in the following three cases: a) the residue of $q$ modulo $m$ generates the multiplicative group of the field of order $m$ (in particular, this holds for $m=3$); b) $m=5$; c) $n=2$.