Base sizes for sporadic simple groups

Let G be a permutation group acting on a set Ω. A subset of Ω is a base for G if its pointwise stabilizer in G is trivial. We write b(G) for the minimal size of a base for G. We determine the precise value of b(G) for every primitive almost simple sporadic group G, with the exception of two cases involving the Baby Monster group. As a corollary, we deduce that b(G) ⩽ 7, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. This settles a conjecture of Cameron.     

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set . A base for Gis a subset B with pointwise stabilizer in G that is trivial;we write b(G) for the smallest size of a base for G. In thispaper we prove that b(G) 6 if G is an almost simple group ofexceptional Lie type and is a primitive faithful G-set. Animportant consequence of this result, when combined with otherrecent work, is that b(G) 7 for any almost simple group G ina non-standard action, proving a conjecture of Cameron. Theproof is probabilistic and uses bounds on fixed point ratios.     

Generation of classical groups over finite fields

Let U_n(V) and Sp_n(V) denote the unitary group and the symplectic group of the n dimensional vector space V over a finite field of characteristic not 2, respectively. Assume that the hyperbolic rank of U_n(V) is at least one. Then U_n(V) is generated by 4 elements and Sp_n(V) by 3 elements. Further, U_2m+1(V) is generated by 3 elements and Sp_4m(V) by 2 elements.     

Conjugacy class sizes and solvability of finite groups

Let G be a finite group and G?? be the set of primary, biprimary and triprimary elements of G. We prove that if the conjugacy class sizes of G?? are {1,m,n,mn} with positive coprime integers m and n, then G is solvable. This extends a recent result of Kong (Manatsh. Math. 168(2) (2012) 267–271).     

Minimal characters of the finite classical groups

Let G(q) be a finite simple group of Lie type over a finite field of order q and d(G(q)) the minimal degree of faithful projective complex representations of G(q). For the case G(q) is a classical group we deter-mine the number of projective complex characters of G(q) of degree d(G(q)). In several cases we also determine the projective complex characters of the second and the third lowest degrees. As a corollary of these results we deduce the classification of quasi-simple irreducible complex linear groups of degree at most 2r r a prime divisor of the group order.     

Conjugacy class sizes and solvability of finite groups

Let G be a finite group and let G* be the set of elements of primary, biprimary and triprimary orders of G. We show that suppose that the conjugacy class sizes of G* are exactly {1, p ^a , n, p ^a n} with (p, n)?=?1 and a??? 0, then G is solvable.     

Identities arising from Gauss sums for finite classical groups

We consider Gauss sums for various finite classical groups, combine our previous results about explicit expressions for those sums with new ones obtained from our main formula based on Deligne-Lusztig theory and get some interesting identities, which are of combinatorial nature and involve various classical exponential sums.     

Characterization of classical groups by orbit sizes on the natural module

We show that if is a finite vector space, and is a subgroup of having the same orbit sizes on 1-spaces as an orthogonal or unitary group on , then, with a few exceptions, is itself an orthogonal or unitary group on .

     

Rational invariants of certain classical groups over finite fields

Let be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over and be the rational function field over . We seek to understand the structure of the rational invariant subfield . In this paper, we prove that is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.