The Double Cosets of a Finite Group
نویسنده
چکیده
1. Introduction. It is the purpose of this paper to study some of the properties of the double cosets of a finite group and to prove two main theorems which generalize the results of two previous papers by the author, 2 giving some relations between the double cosets and the irreducible components of the permutation group generated by a given subgroup. We let H be an arbitrary but fixed subgroup of order h of a finite group G of order g, g — nh, and we let GH be the permutation group of degree n induced by right multiplication of the cosets HSi, i = l, 2, • • • , n, by elements of G. When written as a group of permutation matrices and completely reduced, the group G H will have r' distinct irreducible components I\ of degree ni and multi-plicity ixf, and we may write
منابع مشابه
Enumeration of Double Cosets
Let H and K be subgroups of a group G. The double cosets of H and K in G are the sets HgK, g E G. In this paper we describe a procedure, P, for determining the cardinality of the set H\G/K of double cosets of H and K in G given a finite presentation for G and finite sets of generators for H and K. It is well known that the problem of determining whether or not a group G defined by a finite pres...
متن کاملA Remark on Double Cosets
If a soluble group G contains two finitely generated abelian subgroups A, B such that the number of double cosets AgB is finite, then G is shown to be virtually polycyclic.
متن کاملRepresentations of Double Coset Lie Hypergroups
We study the double cosets of a Lie group by a compact Lie subgroup. We show that a Weil formula holds for double coset Lie hypergroups and show that certain representations of the Lie group lift to representations of the double coset Lie hypergroup. We characterize smooth (analytic) vectors of these lifted representations.
متن کاملParabolic Double Cosets in Coxeter Groups
Parabolic subgroups WI of Coxeter systems (W,S), as well as their ordinary and double quotients W/WI and WI\W/WJ , appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this article...
متن کاملMinimal Length Elements in Some Double Cosets of Coxeter Groups
We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig’s G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [GKP].
متن کامل