The spread of a group G, written s(G), is the largest k such that for any nontrivial elements x1,…,xk∈G there exists y∈G G=〈xi,y〉 all i. Burness, Guralnick and Harper recently classified finite groups G s(G)>0, which involved reduction to almost simple groups. In this paper, we prove an asymptotic result determines exactly when s(Gn)→∞ sequence (Gn). We apply probabilistic geometric ideas, but ...

The symmetries described by Pin groups are the result of combining a finite number discrete reflections in (hyper)planes. current work shows how an analysis using geometric algebra provides picture complementary to that classic matrix Lie approach, while retaining information about given transformation. This imposes type graded structure on groups, not evident their representation. Embracing th...

We prove a myriad of results related to the stabilizer in an algebraic group G generic vector representation V over algebraically closed field k. Our are on level schemes, which carries more information than considering both Lie algebra and G(k) k-points. For simple faithful irreducible, we existence general position, sometimes called principal orbit type. determine those for position is smooth...

Affine difference algebraic groups are a generalization of affine obtained by replacing equations with equations. We show that the isomorphism theorems from abstract group theory have meaningful analogs for these and we establish Jordan–Hölder type theorem allows us to decompose any into almost-simple groups. also characterize via