We explore transversals of finite index subgroups of finitely generated groups. We show that when H is a subgroup of a rank-n group G and H has index at least n in G, we can construct a left transversal for H which contains a generating set of size n for G; this construction is algorithmic when G is finitely presented. We also show that, in the case where G has rank n ≤ 3, there is a simultaneous left-right transversal for H which contains a generating set of size n for G. We finish by showing that if H is a subgroup of a rank-n group G with index less than 3 · 2 n−1 , and H contains no primitive elements of G, then H is normal in G and G/H C n 2 .2010 Mathematics subject classification: primary 20F05; secondary 20E99.
Abstract. Let H, K be subgroups of G. We investigate the intersection properties of left and right cosets of these subgroups.If H and K are subgroups of G, then G can be partitioned as the disjoint union of all left cosets of H, as well as the disjoint union of all right cosets of K. But how do these two partitions of G intersect each other? Definition 1. Let G be a group, and H a subgroup of G. A left transversal for H in G is a set {t α } α∈I ⊆ G such that for each left coset gH there is precisely one α ∈ I satisfying t α H = gH. A right transversal for H in G in defined in an analogous fashion. A left-right transversal for H is a set S which is simultaneously a left transversal, and a right transversal, for H in G.A useful tool for studying the way left and right cosets interact, and obtaining transversals, is the coset intersection graph which we introduce here.Definition 2. Let G be a group and H, K subgroups of G. We define the coset intersection graph Γ G H,K to be a graph with vertex set consisting of all left cosets of H ({l i H} i∈I ) together with all right cosets of K ({Kr j } j∈J ), where I, J are index sets. If a left coset of H and right coset of K correspond, they are still included twice. Edges (undirected) are included whenever any two of these cosets intersect, and the edge aH − Kb corresponds to the set aH ∩ Kb.Observing that left (respectively, right) cosets do not intersect, we see that Γ G H,K is a bipartite graph, split between {l i H} i∈I and {Kr j } j∈J .For H a finite index subgroup of G, the existence of a left-right transversal is well known, sometimes presented as an application of Hall's marriage theorem [3]. When G is finite H will have size n, so any set of k left cosets of H intersects at least k right cosets of H (or their union would have size < kn).
In this paper we introduce a new technique based on high-dimensional Chebyshev tensors, called the orthogonal Chebyshev sliding technique. We implemented this technique inside the systems of a tier-one bank to approximate front-office pricing functions with the aim of reducing the substantial computational burden associated with the FRTB-IMA capital calculation. In all cases, the computational burden reductions obtained were of more than 90 percent, while keeping high degrees of accuracy. The latter obtained as a result of the mathematical properties enjoyed by Chebyshev tensors.
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