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 (?)) not satisfying a particular subgroup property in a particular group or type of group (namely, Symmetric group:S3 (?)).
Let be the subgroup . In other words, is the symmetric group of degree two embedded inside .
Then, is not a normal subgroup of .
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 .
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.
Consider the element and . The commutator is . Thus, is not normal in .