Cosets of symmetric groups

Question

What is an element from each left coset of s6 in s7?

I don't completely understand cosets in symmetric groups.

Answer 1

The way to think of cosets is as a set that the group acts transitively on, with the subgroup as a point stabilizer. If you think about it, S6 inside S7 stabilizes a point in the natural action on 7 letters. If you choose S6 to stabilize 1 (i.e., it only acts on the numbers 2 through 7) then the right cosets correspond to the sets of all elements mapping 1 to 1, mapping 1 to 2, ..., mapping 1 to 7, and the left cosets correspond to the sets of all elements mapping 1 to 1, 2 to 1, ..., 7 to 1.

Answer 2

Lets work with smaller symmetric groups. Let's let $G=S_3$, $H=S_2$. We can actually write out each group in set notation;

$$S_3=\{(1), (12), (13), (23), (123), (132)\}$$

$$S_2=\{(1), (12)\}$$

Now, for any $g\in S_3$, a left coset is a set $gH$. What does that mean? Well, lets look at $g=(23)$. We can "multiply" all elements in $S_2$ by $(23)$. Of course, when working with the symmetric group, we are composing, not multiplying, but you get the idea i hope. Then

$$(23)H=(23) \{(1),(12)\}=\{(23)(1), (23)(12)\}=\{(23),(132)\}$$

This is one of the left cosets of $H=S_2$ in $S_3$. The same principle can be applied to your problem, but of course there are a lot more elements in $S_6$ and $S_7$, so now using @C Monsour's answer should help to enlighten a bit more...