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