List all left and right cossets of S3 in S4SolutionWe consid

List all left and right cossets of S₃ in S_4?

Solution

We consider the subgroup H in the group G defined as follows.

G is the symmetric group of degree four, which, for concreteness, we take as the symmetric group on the set {1,2,3,4}.

H is the subgroup of G comprising those permutations that fix {4}.

In particular, H is the symmetric group on {1,2,3}, embedded naturally in G.

It is isomorphic to symmetric group:S3. H has order 6.

There are three other conjugate subgroups to H in G (so the total conjugacy class size of subgroups is {4). The other subgroups are the subgroups fixing {1}, {2}, and {3} respectively.

The four conjugates are:

H=H₄={(),(1,2),(1,3),(2,3),(1,2,3),(1,3,2)}

H₁={(),(2,3),(3,4),(2,4),(2,3,4),(2,4,3)}

H₂={(),(1,3),(3,4),(1,4),(1,3,4),(1,4,3)}

H₃= {(),(1,2)(2,4)(1,4)(1,2,4)(1,4,2)}

There are four left cosets and four right cosets of each subgroup. Each left coset of a subgroup is a right coset of one of its conjugate subgroups. This gives a total of 16 subsets.

The cosets are parametrized by ordered pairs (i,j){1,2,3,4} X {1,2,3,4}. The coset parametrized by (i,j) is the set of all elements that send i to j. This is a left coset of H_i and a right coset of H_j.