Invariant theory of finite general linear groups modulo Frobenius powers

GLn(Fq) in polynomials modulo Frobenius powers

Invariant Theory of Finite Groups

This lecture is concerned with polynomial invariants of finite groups which come from a linear group action. We introduce the basic notions of commutative algebra needed and discuss structural properties of invariant rings.

Invariants of GLn(Fq) in polynomials modulo Frobenius powers

Powers in finite groups

If G is a finitely generated profinite group then the verbal subgroup Gq is open. In a d-generator finite group every product of qth powers is a product of f(d, q) qth powers. 20E20, 20F20.

Sieve Methods in Group Theory I: Powers in Linear Groups

The sieve method is a classic one in number theory (see, for example, [FI]). Recently it found some applications in a non-commutative setting. On the one hand, Bourgain-Gamburd-Sarnak [BGS1] applied it in studying almost-prime vectors in orbits of non-commutative groups acting on Z. On the other hand, Rivin [Ri] and Kowalski [Ko] used it to study generic properties of elements in the mapping cl...