On Short Expressions for Cosets of Permutation Subgroups

Automatic groups, subgroups and cosets

The history, definition and principal properties of automatic groups and their generalisations to subgroups and cosets are reviewed briefly, mainly from a computational perspective. A result about the asynchronous automaticity of an HNN extension is then proved and applied to an example that was proposed by Mark Sapir. AMS Classification 20F32; 20F05

A Note on Short Cosets

We describe some improved techniques for the computation of short cosets. This speeds up the computation of Galois groups in magma.

Cosets, Lagrange’s theorem and normal subgroups

Our goal will be to generalize the construction of the group Z/nZ. The idea there was to start with the group Z and the subgroup nZ = 〈n〉, where n ∈ N, and to construct a set Z/nZ which then turned out to be a group (under addition) as well. (There are two binary operations + and · on Z, but Z is just a group under addition. Thus, the fact that we can also define multiplication on Z/nZ will not...

A Model for the Double Cosets of Young Subgroups

There are several results concerning double cosets of Young subgroups of symmetric groups which are often proved as corollaries of more general results about Coxeter groups. In this note we give a selection of these results and give them the short combinatorial proofs they deserve. Our replacement for the machinery of Coxeter groups consists of some simple combinatorial arguments involving tabl...

MATH 436 Notes: Subgroups and Cosets