arXiv subjects
Publications and source records attributed to Alexey M. Staroletov.
For a finite group $G$, let $ω(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $ω(H)=ω(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if it is finite, then all finite groups $H$ with $ω(H)=ω(G)$ are listed. We complete the solution of the recognition problem for all finite simple groups.
For a finite group $G$, let $ω(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $ω(H)=ω(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if $h(G)$ is finite, then all finite groups $H$ with $ω(H)=ω(G)$ are described. We complete the solution of the recognition problem for the finite simple linear and unitary groups.
The Gruenberg-Kegel graph $Γ(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of order $rs$ in $G$. A finite group $G$ is called almost recognizable (by Gruenberg-Kegel graph) if there is only finite number of pairwise non-isomorphic finite groups having Gruenberg-Kegel graph as $G$. If $G$ is not almost recognizable, then it is called unrecognizable (by Gruenberg--Kegel graph). Recently P.J. Cameron and the first author have proved that if a finite group is almost recognizable, then the group is almost simple. Thus, the question of which almost simple groups (in particular, finite simple groups) are almost recognizable is of prime interest. We prove that every finite simple exceptional group of Lie type, which is isomorphic to neither ${^2}B_2(2^{2n+1})$ with $n\geq1$ nor $G_2(3)$ and whose Gruenberg-Kegel graph has at least three connected components, is almost recognizable. Moreover, groups $ {^2}B_2(2^{2n+1})$, where $n\geq1$, and $G_2(3)$ are unrecognizable.
The spectrum of a finite group is the set of its elements orders. Groups are said to be isospectral if their spectra coincide. For every finite simple exceptional group $L=E_7(q)$, we prove that each finite group isospectral to $L$ is isomorphic to a group $G$ squeezed between $L$ and its automorphism group, that is $L\leq G\leq \operatorname{Aut}L$; in particular, up-to isomorphism, there are only finitely many such groups. This assertion, together with a series of previously obtained results, implies that the same is true for every finite simple exceptional group except the group ${}^3D_4(2)$.