# S2 is not normal in S3

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Cyclic group:Z2 (?))notsatisfying a particular subgroup property in a particular group or type of group (namely, Symmetric group:S3 (?)).

## Statement

Let be the symmetric group:S3, which is the group of all permutations on . The group has the following six elements, given by their cycle decompositions:

Let be the subgroup . In other words, is the symmetric group of degree two embedded inside .

Then, is *not* a normal subgroup of .

To explore more, see element structure of symmetric group:S3 and subgroup structure of symmetric group:S3.

## Proof

### Using conjugation

Consider the element and its action by conjugation on . Since has order two, it equals its inverse, so . The element is not in . Thus, is not normal in .

### Using cosets

`Further information: S2 in S3#Cosets`

The left cosets of in are:

The right cosets of in are:

We see that the space of left cosets does not match the space of right cosets.

### Using commutators

Consider the element and . The commutator is . Thus, is not normal in .