One of the simplest and most useful notions in mathematics is that of a group action: if G is a group and X is a nonempty set, then an action of G o n X or a G-set structure on X consists of a multiplication operation G X ! X, with the image of a pair g;x written as, say, gx, with the following axioms satissed: 1 1x = x for all x 2 X here 1 2 G is the identity element o f G; 2 ghx = ghx for all...

We show that any non-zero orbit under a non-compact, simple, irreducible linear group is dense in the Bohr compactification of the ambient space.

Let G be a finite group and V be a finite G–module. We present upper bounds for the cardinalities of certain subsets of Irr(GV ), such as the set of those χ ∈ Irr(GV ) such that, for a fixed v ∈ V , the restriction of χ to 〈v〉 is not a multiple of the regular character of 〈v〉. These results might be useful in attacking the non–coprime k(GV )–problem.

We develop tools for selective inference in the setting of group sparsity, including the construction of confidence intervals and p-values for testing selected groups of variables. Our main technical result gives the precise distribution of the magnitude of the projection of the data onto a given subspace, and enables us to develop inference procedures for a broad class of group-sparse selectio...