Let $G = S_n$, fix $i \in \{1,...,n\}$ and let $G_i = \{\sigma \in G ~|~ \sigma(i) = i\}$ .Then find $|G_i|?$.
My attempt :I think $|G_i|= |S_n|$ because $G_i$ permutes all n elements of the set $\{1,2,3,...,n\}$
For example take $S_3= G,$ $G_i = \{\sigma \in S_3 ~|~ \sigma(i) = i\}$
Fix $i= \{1,2,3\}$.Element of $S_3$ are $\{1,(12),(13),(23),(123),(132)\}$
For $\sigma=(1)$
$G_1 = \{\sigma \in S_3 ~|~ (1)(1) = 1\}$
$G_2 = \{\sigma \in S_3 ~|~ (1)(2) = 2\}$
$G_3 = \{\sigma \in S_3 ~|~ (1)(3) = 3\}$
For $\sigma=(12)$
$G_1 = \{\sigma \in S_3 ~|~ (12)(1) = 1\}$
$G_2 = \{\sigma \in S_3 ~|~ (12)(2) = 2\}$
$G_3 = \{\sigma \in S_3 ~|~ (12)(3) = 3\}$
For $\sigma=(13)$
$G_1 = \{\sigma \in S_3 ~|~ (13)(1) = 1\}$
$G_2 = \{\sigma \in S_3 ~|~ (13)(2) = 2\}$
$G_3 = \{\sigma \in S_3 ~|~ (13)(3) = 3\}$
For $\sigma=(23)$
$G_1 = \{\sigma \in S_3 ~|~ (23)(1) = 1\}$
$G_2 = \{\sigma \in S_3 ~|~ (23)(2) = 2\}$
$G_3 = \{\sigma \in S_3 ~|~ (23)(3) = 3\}$
For $\sigma=(123)$
$G_1 = \{\sigma \in S_3 ~|~ (123)(1) = 1\}$
$G_2 = \{\sigma \in S_3 ~|~ (123)(2) = 2\}$
$G_3 = \{\sigma \in S_3 ~|~ (123)(3) = 3\}$
For $\sigma=(132)$
$G_1 = \{\sigma \in S_3 ~|~ (132)(1) = 1\}$
$G_2 = \{\sigma \in S_3 ~|~ (132)(2) = 2\}$
$G_3 = \{\sigma \in S_3 ~|~ (132)(3) = 3\}$
For fixed $i=1,2,3$ we have $|G_i|=3+3+3+3+3+3=18 $