Bases of quasisimple linear groups

Let $V$ be a vector space of dimension $d$ over $F_q$, a finite field of $q$ elements, and let $G \le GL(V) \cong GL_d(q)$ be a linear group. A base of $G$ is a set of vectors whose pointwise stabiliser in $G$ is trivial. We prove that if $G$ is a quasisimple group (i.e. $G$ is perfect and $G/Z(G)$ is simple) acting irreducibly on $V$, then excluding two natural families, $G$ has a base of size at most 6. The two families consist of alternating groups ${\rm Alt}_m$ acting on the natural module of dimension $d = m-1$ or $m-2$, and classical groups with natural module of dimension $d$ over subfields of $F_q$

Finite subgroups of simple algebraic groups with irreducible centralizers

We determine all finite subgroups of simple algebraic groups that have irreducible centralizers - that is, centralizers whose connected component does not lie in a parabolic subgroup.Comment: 24 page

Recognition of finite exceptional groups of Lie type

Let $q$ be a prime power and let $G$ be an absolutely irreducible subgroup of $GL_d(F)$, where $F$ is a finite field of the same characteristic as \F_q, the field of $q$ elements. Assume that $G \cong G(q)$, a quasisimple group of exceptional Lie type over \F_q which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from $G$ to the standard copy of $G(q)$. If $G \not\cong {}^3 D_4(q)$ with $q$ even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle

The length and depth of algebraic groups

Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic group of rank $r$ has length $\dim B + r$, which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group $G$ exceeds $\frac{1}{2} \dim G$. We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most $6$ (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group $G$, the dimension of $G/R(G)$ is bounded above in terms of the chain difference of $G$.Comment: 18 pages; to appear in Math.

Intersections of Matrix Algebras and Permutation Representations of PSL(n,q)

AbstractIf G is a group, H a subgroup of G, and Ω a transitive G-set we ask under what conditions one can guarantee that H has a regular orbit (=of size |H|) on Ω. Here we prove that if PSL(n,q)⊆G⊆PGL(n,q) and H is cyclic then H has a regular orbit in every non-trivial G-set (with few exceptions). This result is obtained via a mixture of group theoretical and ring theoretical methods: Let R be the ring of all n×n matrices over the finite field F and let Z be the subring of scalar matrices. We show that if A and M are proper subrings of R containing Z, and if A is commutative and semisimple, then there exists an element x∈SL(n,F) such that xAx−1∩M=Z or n=2=|F|