Left cosets, right cosets & Lagrange's theorem

Question

Consider the group $S_3$ and the cyclic subgroups $G:=\langle(1 2 3)\rangle$ and $H:=\langle(3 1)\rangle$. how would you find left and then right cosets of these subgroups as subsets of s3?

How would these answers support Lagrange's theorem?

Answer 1

First, let's recall the definition of a coset:

Let $G$ be a group. The left cosets of $H<G$ are all the equivalence classes of the relation $(\sim)$ where $g_1\sim g_2$ if $g_1 = g_2h$ for some $h\in H$.

So, to find the left cosets of $H=\langle(123)\rangle$, we multiply elements of $S_3$ on the right by elements of $H$ to find our equivalence classes. For example, if $\sigma\in S_3$, then $\{\sigma, \sigma(123), \sigma(132)\}$ would be a coset of $H$. Do this for each element in $S_3$, and look at the sets you have. Hint: some of them should be the same!

To show how this might provide evidence to Lagrange's Theorem, consider that equivalence classes of a set $S$ are a partition of $S$, and look at the size of each of your cosets. Are they different? Are they the same? How do they relate to the size of $S_3$ and the size of $H$?