Proportions of r -regular elements in finite classical groups

Regular Elements of Finite Reflection Groups

Regular Elements and Monodromy of Discriminants of Finite Reflection Groups

where the product is over all reflection hyperplanes of G. Let A : V/G -+ @ be the map induced by 6, thus A is the discriminant of G. A subgroup of G is called parabolic if it is generated by all reflections of G fixing elementwise a given subspace of V. The degrees of G are denoted by dl , d2, . . . , d,,. We call a degree of G primitive if it is bigger than the degrees of all proper parabolic...

Pairwise‎ ‎non-commuting elements in finite metacyclic $2$-groups and some finite $p$-groups

Let $G$ be a finite group‎. ‎A subset $X$ of $G$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $X$ do not commute‎. ‎In this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

On the number of p-regular elements in finite simple groups

A p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n− 1 is the dimens...

Eigenvalues of unipotent elements in cross-characteristic representations of finite classical groups

Let H be a finite classical group, g be a unipotent element of H of order s and θ be an irreducible representation of H with dim θ > 1 over an algebraically closed field of characteristic coprime to s. We show that almost always all the s-roots of unity occur as eigenvalues of θ(g), and classify all the triples (H,g, θ) for which this does not hold. In particular, we list the triples for which ...