Let [Formula: see text] be a subgroup of the full automorphism group of a [Formula: see text]-[Formula: see text] symmetric design [Formula: see text]. If [Formula: see text] is flag-transitive and point-primitive, then Soc[Formula: see text] cannot be [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].
AbstractThis paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix M of size n over a field {\mathbb{F}} with the property that there exists {\alpha\in\mathbb{F}} such that M is similar to {\operatorname{diag}(\alpha\cdot\mathrm{Id}_{k},M_{1})}, where {M_{1}} is cyclic and {0\leq k\leq n}).
While a previous paper dealt with the Weil representations of finite classical groups, which play a key role in the general picture, the present paper provides a conclusive answer for all cross-characteristic projective irreducible representations of the finite quasi-simple groups of Lie type and their automorphism groups.
The degree pattern of a finite group G is introduced in [10] and it is proved that the following simple groups are uniquely determined by their degree patterns and orders: all sporadic simple groups, alternating groups Ap (p ≥ 5 is a twin prime) and some simple groups of Lie type. In this paper, we continue this investigation. In particular, we show that the automorphism groups of sporadic simple groups (except Aut (J2) and Aut (McL)), all simple C22-groups, the alternating groups Ap, Ap+1, Ap+2 and the symmetric groups Sp, Sp+1, where p is a prime, are also uniquely determined by their degree patterns and orders.
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 $(45,12,3)$.
Almost cyclic elements in cross-characteristic representations of finite groups of Lie type