On Pairs of Additive Subgroups Associated with Intermediate Subgroups of Groups of Lie Type over Nonperfect Fields

The author has previously (Trudy IMM UrO RAN, 19(2013), no. 3) described the groups lying between twisted Chevalley groups G(K) and G(F) of type 2Al, 2Dl, 2E6, 3D4 in the case when the larger field F is an algebraic extension of the smaller nonperfect field K of exceptional characteristic for the group G(F) (characteristics 2 and 3 for the type 3D4 and only 2 for other types). It turned out that apart from, perhaps, the type 2Dl, such intermediate subgroups are standard, that is, they are exhausted by the groups G(P)H for some intermediate subfield P, K ⊆ P ⊆ F, and of the diagonal subgroup H normalizing the group G(P). In this note, it is established that intermediate subgroups are also standard for the type 2Dl

Commuting Involution Graphs for Certain Exceptional Groups of Lie Type

Suppose that G is a finite group and X is a G-conjugacy classes of involutions. The commuting involution graph $${\mathcal {C}}(G,X)$$ C ( G , X ) is the graph whose vertex set is X with $$x, y \in X$$ x , y ∈ X being joined if $$x \ne y$$ x ≠ y and $$xy = yx$$ x y = y x . Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs.

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 $(45,12,3)$.

THE INDUCTIVE BLOCKWISE ALPERIN WEIGHT CONDITION FOR TYPE AND THE PRIME

We establish the inductive blockwise Alperin weight condition for simple groups of Lie type $\mathsf C$ and the bad prime $2$ . As a main step, we derive a labelling set for the irreducible $2$ -Brauer characters of the finite symplectic groups $\operatorname {Sp}_{2n}(q)$ (with odd q), together with the action of automorphisms. As a further important ingredient, we prove a Jordan decomposition for weights.