Let $G$ be a finite group. Denote by $\textrm{Irr}(G)$ the set of all irreducible complex characters of $G.$ Let $\textrm{cd}(G)=\{χ(1)\;|\;χ\in \textrm{Irr}(G)\}$ be the set of all irreducible complex character degrees of $G$ forgetting multiplicities, and let $\textrm{X}_1(G)$ be the set of all irreducible complex character degrees of $G$ counting multiplicities. Let $H$ be any non-abelian simple exceptional group of Lie type. In this paper, we will show that if $S$ is a non-abelian simple group and $\textrm{cd}(S)\subseteq \textrm{cd}(H)$ then $S$ must be isomorphic to $H.$ As a consequence, we show that if $G$ is a finite group with $\textrm{X}_1(G)\subseteq \textrm{X}_1(H)$ then $G$ is isomorphic to $H.$ In particular, this implies that the simple exceptional groups of Lie type are uniquely determined by the structure of their complex group algebras.
The paper studies the minimum polynomial degrees of $p$-elements in cross-characteristic representations of simple groups of exceptional Lie type whose BN-pair rank is at most 2. Specifically, we prove that the degree in question equals the order of the element.
In this article, we study symmetric $(v, k, λ)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle $X$ is a finite simple exceptional group of Lie type. We prove a reduction theorem, severely restricting the possible parameters of such designs. We also prove that the parameters $k$ and $λ$ are not coprime, and neither of these parameters can be prime. Moreover, if $λ$ is at most $100$, we show that there are two such parameters sets, namely, $(351,126,45)$ and $(378,117,36)$ for $G=X=G_{2}(3)$. Our analysis depends heavily on detailed information about a…
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)…
We study embeddings of $\mathrm{PSL}_2(p^a)$ into exceptional groups $G(p^b)$ for $G=F_4,E_6,{}^2\!E_6,E_7$, and $p$ a prime with $a,b$ positive integers. With a few possible exceptions, we prove that any almost simple group with socle $\mathrm{PSL}_2(p^a)$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$ and $E_7$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type $A_1$ inside the algebraic group. Together with a recent result of Burness and Testerman for $p$ the Coxeter number plus one, this proves t…
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 gr…
