# Symmetric group on a finite set is a Coxeter group

This article states and proves a fact about a particular group, or kind of group, (i.e., Symmetric group (?)) having a particular presentation (or kind of presentation), i.e., where the generic name for such presentations or groups having such presentations is: Coxeter presentation (?)

View other facts about presentations for particular groups

## Contents

## Statement

Suppose is a nonnegative integer and is the symmetric group on the set . Then, is isomorphic to a Coxeter group with generators , where and for and differing by more than one.

In other words:

.

The isomorphism identifies with the transposition .

## Facts used

## Proof

### The Coxeter group described admits a surjective homomorphism to the symmetric group

Consider a map:

given by:

.

To check that the map is well-defined, we need to check that all the Coxeter relations are satisfied by the images of the . This is direct: the square of any transposition is the identity element, the product of two adjacent transpositions has order three, and the product of two disjoint transpositions has order two.

Next, the map is surjective because by fact (1), elements of the form generate .

### The size of the Coxeter group is at most

Let be the Coxeter group for letters.

First observe that the subgroup of generated by satisfies all the relations for the Coxeter group for generators, hence is a homomorphic image of . Hence, by induction, .

We now show that has at most left cosets in . In fact, we can show that the every left coset of in has a representative among these:

.

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Thus, . Using Lagrange's theorem, we get .