One of my exercises in my textbook is:

Find all left cosets of $\langle(12), (34)\rangle$ in $S_4$.

Note: $\langle x,y,\dots\rangle$ denotes the smallest subgroup containing $\{x,y,\dots\}.$

I'm not sure how to represent $H$ because of the note. Does $H = \langle (12)\rangle$ and $\langle(34)\rangle$? So, $H = \{ e, (12), (34)\}$? Which mean $| G | / | H | = 24/3 = 8$ left cosets? Then, I just need to find my cosets. I'm just not sure about the value of my $H.$

Thank you,

## Best Answer

Note that $H=\{ e, (12), (34), (12)(34)\}$. There are $|S_4|/|H|=4!/4=6$ left cosets.

To find the left cosets of a subgroup $K$ of a group $G$, recall that

$$aK=\{ ak\mid k\in K\}$$

for each $a\in G$. All you need to do, then, is multiply each element of $H$ on the left by each element of $S_4$, and see which are equal.