How to derive the cosets of $A_4$?

Question

In the alternating group $A_4$ for the subgroup of $H\le A_4$ find all of its left and right cosets where $$H=\{\text{Id},(1,2,3),(1,3,2)\}.$$

We have by Lagrange's theorem that $$|A_4:H|=\frac{|A_4|}{|H|}=\frac{24/2}{3}=4$$ which means that $H$ has $4$ left and $4$ right cosets.

By definition a left coset of $H$ in $A_4$ determined by $x\in A_4$ is the set $\{xh:h\in H\}$.

So, the left cosets will be $x\text{Id},x(1,2,3),x(1,3,2)$, right?

How can I find the $4$th coset?

Answer 1

First note that $$A_4=\{\text{Id},(1,2,3),(1,3,2),(2,4,3),(2,3,4),(1,2)(3,4),(1,2,4),(1,3,4),(1,3)(2,4),(1,4,2),(1,4,3),(1,4)(2,3)\}$$ We have, \begin{align} (1,2)(3,4)\cdot(1,2,3)=(2,4,3)\\ (1,3)(2,4)\cdot(1,2,3)=(1,4,2)\\ (1,4)(2,3)\cdot(1,2,3)=(1,3,4) \end{align} and \begin{align} (1,2)(3,4)\cdot(1,3,2)=(1,4,3)\\ (1,3)(2,4)\cdot(1,3,2)=(2,3,4)\\ (1,4)(2,3)\cdot(1,3,2)=(1,2,4) \end{align} So the left cosets of $H$ are $\text{Id}H,(1,2)(3,4)H,(1,3)(2,4)H,(1,4)(2,3)H$.

We have to repeat the above calculations for $Hx$ (right coset).