Let $G_i=\{\sigma\in S_n:\sigma(i)=i\}$. Describe the cosets of $\frac{S_n}{G_n}$. How many cosets are there? List them.

Question

Let $S_n$ be the symmetric group of permutations of ${1,2,..., n}$ and let $i = {1,2,...,n}$.

Let $G_i = \{ \sigma \in S_n : \sigma(i) =i\}$. Describe the cosets of $\frac{S_n}{G_n}$. How many cosets are there? List them.

Edit : I could prove that $G_i$ is a subgroup, even a normal subgroup which was also given in the problem.

$$\text{Let,} \alpha , \beta \in G_i \\ \implies \alpha(i) = i , \beta (i) = i \\ \implies \beta^{-1}(i) = i \\ \implies \alpha (\beta^{-1}(i)) = \alpha(i) = i \\ \implies \alpha \beta^{-1} \in G_i $$ Therefore $G_i$ is a subgroup of $S_n$.

Also for any $f \in S_n$ and any $\sigma \in G_i$

$$ f \sigma (f^{-1}(i)) = f(f^{-1})(i) = i \\ \implies f \sigma f^{-1} \in G_i $$

Therefore, $G_i$ is normal in $S_n$.

But for the rest of it I couldn't go any further.

Any suggestions/hint where to start, I'm totally lost.

Answer 1

When we think of cosets, we think of an equivalence relation. What is an equivalence relation? It is a way of grouping together objects that we deem as equivalent. Suppose I ask you the following question, how many different groups of triangles are there such that two triangles having the same angles are grouped together? You'd pick a triangle out of each group and there would be as many triangles as there are 3-multisets $\lbrace a,b,\pi-a-b\rbrace$ with all elements positive.

The question you want the answer to is this. How many different types of permutations are there if I put two permutations with the same value for $n$ together? You'd put the permutations with $\sigma(n)=1$ in a group, ones with $\sigma(n)=2$ in a different group and so on. How many groups are there?

Look at the set of $\sigma \in S_n$ such that $\sigma(n)=i$ for a fixed $i$, with $1\leq i \leq n$. Call these sets $H_i$. How does one go from $G_n$ to a set $H_i$?

Also, can you see that $\cup H_i=S_n$? Every permutation must take the element $n$ somewhere and these sets account for all of them. What are the cardinalities of these $H_i$? What is $H_i \cap H_j$?

Answer 2

First of all notice that $|G_n| = (n-1)!$. From Lagrange we get that $\left|\frac{S_n}{G_n}\right|= \frac{|S_n|}{|G_n|} = n$ so there are $n$ cosets. Now consider for each $k=1,\dots, n$ the permutation $\sigma_k = (kn)$. Next show that $\sigma_k G_n \cap \sigma_l G_n = \emptyset$ for $k\neq l$ which forces the set $\{\sigma_k G_n: k=1,\dots, n\}$ to be the set of cosets of $G_n$.