Let H be the cyclic subgroup of the alternating group A4 gen

Let H be the cyclic subgroup of the alternating group A_4 generated by the permutation (123). Exhibit the left and the right cosets of H explicitly.

Solution

The number of left or right cosets = order of A₄ / order of H = 12/3 = 4

Select P a subgroup of A₄ having 4 elements: {I , (1 2)(3 4) , (1 3)(2 4), (1 4)(2 3)}

For left cosets pair H on the left side with P

Left coset of H = IH , (1 2)(3 4)H , (1 3)(2 4)H , (1 4)(2 3)H

We should check that none of these cosets duplicate others,

We observe that none of (12)(34)H, (13)(24)H, (14)(23)H equals H as each contains an element of P which is not in H.

Similarly to fnd right cosets of H we pair it on the right side of P.

Right cosets of H = H I , H(1 2)(3 4) , H(1 3)(2 4) , H(1 4)(2 3)