Curves with prescribed symmetry and associated representations of mapping class groups

Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map from the group algebra $${{\mathbb {Q}}}G$$ Q G to the algebra of $${{\mathbb {Q}}}$$ Q -endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation often acts $${{\mathbb {Q}}}$$ Q -irreducibly in a G-isogeny space of $$H^1(C; {{\mathbb {Q}}})$$ H 1 ( C ; Q ) and with image a $${{\mathbb {Q}}}$$ Q -almost simple group.

Almost Simple Groups of Lie Type and Symmetric Designs with $\lambda$ Prime

In this article, we investigate symmetric $(v,k,\lambda)$ designs $\mathcal{D}$ with $\lambda$ prime admitting flag-transitive and point-primitive automorphism groups $G$. We prove that if $G$ is an almost simple group with socle a finite simple group of Lie type, then $\mathcal{D}$ is either the point-hyperplane design of a projective space $\mathrm{PG}_{n-1}(q)$, or it is of parameters $(7,4,2)$, $(11,5,2)$, $(11,6,3)$ or $(45,12,3)$.

Flag-Transitive Point-Primitive Symmetric $(v,k,\lambda)$ Designs with Large $k$

In 2012, Tian and Zhou conjectured that a flag-transitive and point-primitive automorphism group of a symmetric $(v,k,\lambda)$ design must be an affine or almost simple group. In this paper, we study this conjecture and prove that if $k\leqslant 10^3$ and $G\leqslant Aut(\mathcal{D})$ is flag-transitive, point-primitive, then $G$ is affine or almost simple. This support the conjecture.

Non-solvable groups each of whose character degrees has at most two prime divisors

In this paper, we determine all almost simple groups each of whose character degrees has at most two distinct prime divisors. More generally, we show that a finite non-solvable group [Formula: see text] with this property is an extension of an almost simple group [Formula: see text] by a solvable group and [Formula: see text], where [Formula: see text] is the set of all primes dividing some character degree of [Formula: see text].

GENERATING MAXIMAL SUBGROUPS OF FINITE ALMOST SIMPLE GROUPS

For a finite group $G$ , let $d(G)$ denote the minimal number of elements required to generate $G$ . In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

Non-commuting graphs of certain almost simple groups

Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

Almost simple groups with no product of two primes dividing three character degrees

Let {\operatorname{Irr}(G)} denote the set of complex irreducible characters of a finite group G, and let {\operatorname{cd}(G)} be the set of degrees of the members of {\operatorname{Irr}(G)} . For positive integers k and l, we say that the finite group G has the property {\mathcal{P}^{l}_{k}} if, for any distinct degrees {a_{1},a_{2},\dots,a_{k}\in\operatorname{cd}(G)} , the total number of (not necessarily different) prime divisors of the greatest common divisor {\gcd(a_{1},a_{2},\dots,a_{k})} is at most {l-1} . In this paper, we classify all finite almost simple groups satisfying the property {\mathcal{P}_{3}^{2}} . As a consequence of our classification, we show that if G is an almost simple group satisfying {\mathcal{P}_{3}^{2}} , then {\lvert\operatorname{cd}(G)\rvert\leqslant 8} .

On the prime graph question for almost simple groups with an alternating socle

Let [Formula: see text] be an almost simple group with socle [Formula: see text], the alternating group of degree [Formula: see text]. We prove that there is a unit of order [Formula: see text] in the integral group ring of [Formula: see text] if and only if there is an element of that order in [Formula: see text] provided [Formula: see text] and [Formula: see text] are primes greater than [Formula: see text]. We combine this with some explicit computations to verify the prime graph question for all almost simple groups with socle [Formula: see text] if [Formula: see text].