(1) Let G be a group and H ⊂ Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p2 (p a prime number) is abelian and that the centre of any non-abelian group of order p3 has p elements. (2) (?) Let H be a normal subgroup of a p-group G, H 6= {e}. Prove that H∩Z(G) 6= {e}. In particular, one obtains that any normal subgroup with p-elements is con...

A budget-constrained buyer wants to purchase items from a shortlisted set. Items are differentiated by observable quality and sellers have private reserve prices for their items. The buyer’s problem is to select a subset of maximal quality. Money does not enter the buyer’s objective function, but only his constraints. Sellers quote prices strategically, inducing a knapsack game. We derive the B...

We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply these formulas effectively, the parameter c, which in general is hard to control, is studied in some important situations. These results are then used to provid...

Let (W, I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [BK] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxe...

A semigroup is a nonvoid Hausdorff space together with a continuous associative multiplication, denoted by juxtaposition. In what follows S will denote one such and it will be assumed that S is compact. I t thus entails no loss of generality to suppose that S is contained in a locally convex linear topological space 9C, but no particular imbedding is assumed. For general notions about semigroup...

In this article we formalize the Kuratowski closure-complement result: there is at most 14 distinct sets that one can produce from a given subset A of a topological space T by applying closure and complement operators and that all 14 can be obtained from a suitable subset of R, namely KuratExSet = {1} ∪Q(2, 3) ∪ (3, 4) ∪ (4,∞). The second part of the article deals with the maximal number of dis...

Let I = (a,b) and L be a nowhere dense perfect set containing the ends of the interval I and let φ : I → R be a non-increasing continuous surjection constant on the components of I \ L and the closures of these components be the maximal intervals of constancy of . The family F t t R of the interval-valued functions F t 1 φ { , ∈ } (x): = φ [t + φ(x)], x ∈ I forms a set-valued iteration group. W...

We will give some of this structure theory here: namely, we will classify the maximal compact subgroups of GLn(K) for K a nondiscrete locally compact field. This has immediate applications to the structure of finite subgroups of GLn(Q), which are of intrinsic interest and are quite useful in areas like representation theory and modular and automorphic forms. Moreover, this material (actually, a...

Let G be a semisimple real Lie group of non-compact type, K a maximal compact subgroup and S ⊆ G a semigroup with nonempty interior. We consider the ideal boundary ∂∞(G/K) of the associated symmetric space and the flag manifolds G/PΘ . We prove that the asymptotic image ∂∞(Sx0) ⊆ ∂∞(G/K), where x0 ∈ G/K is any given point, is the maximal invariant control set of S in ∂∞(G/K). Moreover there is ...

Let G be the real points of a complex connected reductive algebraic group GC . Let K be a maximal compact subgroup of G. We parametrize the set K̂ of irreducible representations of K. The goal is to describe an algorithm for such a parametrization and to implement it as a package of the Atlas of Lie groups and representations software developed by Fokko du Cloux.