On coincidence of Gruenberg–Kegel graphs of a finite almost simple group and a finite non-solvable Frobenius group

The Gruenberg–Kegel graph or the prime graph $\Gamma(G)$ of a finite group $G$ is the graph whose vertices are the prime divisors of $|G|$, with $p\sim q$ if and only if $G$ contains an element of order $pq$. Our research is devoted to the problem of describing cases when Gruenberg–Kegel graphs of non-isomorphic finite groups are both disconnected and coincide.

In our investigation we use standard terminology and notations (see [1]). Let's recall some known definitions. A finite group $G$ is a Frobenius group if there is a non-trivial subgroup $C$ of $G$ such that $C \cap gCg^{-1} = \{1\}$ whenever $g \not \in C$, also this subgroup $C$ is called a Frobenius complement of $G$. Let $K=\{1\} \cup (G\setminus \bigcup_{g \in G} gCg^{-1}).$ Then $K$ is a normal subgroup of $G$ which is called the Frobenius kernel of $G$. A finite group $G$ is a $2$-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ and $BC$ are Frobenius groups with kernels $A$ and $B$ and complements $B$ and $C$, respectively. It is known that each $2$-Frobenius group is solvable. The socle $Soc(G)$ of a finite group $G$ is the subgroup of $G$ generated by the set of all its non-trivial minimal normal subgroups. A finite group $G$ is almost simple if $Soc(G)$ is a finite nonabelian simple group.

By the Gruenberg–Kegel Theorem, if $G$ is a finite group with disconnected Gruenberg–Kegel graph, then one of the following holds: $G$ is a Frobenius group, $G$ is a $2$-Frobenius group, $G$ is an extension of a nilpotent group by an almost simple group. In [2] all the cases of coincidence of Gruenberg-Kegel graphs of a finite simple group and of a Frobenius or a $2$-Frobenius group were described. Moreover, directly from the main results of the papers [3] and [4] we can obtain the complete list of almost simple groups such that their Gruenberg-Kegel graphs coincide with Gruenberg-Kegel graphs of solvable Frobenius groups or $2$-Frobenius groups. In this talk we discuss a recent progress in the classification of almost simple (but not simple) groups such that their Gruenberg-Kegel graphs coincide with Gruenberg-Kegel graphs of non-solvable Frobenius groups.

Acknowledgement. This work was supported by the Russian Science Foundation (project 19-71-10067).

This is joint work with Natalia Maslova

Список литературы

1. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press \publadrr Oxford, 1985
2. M. R. Zinov'eva, V. D. Mazurov, “On finite groups with disconnected prime graph”, Proc. Steklov Inst. Math., 283:1 (2013), 139–145
3. I. B. Gorshkov, N. V. Maslova, “Finite almost simple groups whose Gruenberg–Kegel graphs are equal to the Gruenberg–Kegel graphs of solvable groups”, Algebra and Logic, 57:2, 115–129
4. M. R. Zinov'eva, A. S. Kondrat'ev, Finite almost simple groups with prime graphs all of whose connected components are cliques, 295:S1 (2013), 178–188