Hi! It's me again with another counting problem.

I am trying to find the order of the following set (where $\displaystyle n \in \mathbb{Z}$ is fixed): $\displaystyle G_1 = \{ \sigma \in S_n | \sigma (1) = 1 \} $, the stabilizer of 1 in $\displaystyle S_n$.

Now, I have calculated 4 of these:

n = 2: $\displaystyle G_1 = \{ e \}$ so $\displaystyle |G_1| = 1$

n = 3: $\displaystyle G_1 = \{ e, (23) \}$ so $\displaystyle |G_1|= 2$

n = 4: $\displaystyle G_1 = \{ e, (234), (243), (23), (24), (34) \}$ so $\displaystyle |G_1|= 6$

I also did n = 5 which gives: $\displaystyle |G_1| = 24$

The pattern seems to be that $\displaystyle |G_1| = (n - 1)!$, but the examples don't help me to understand the counting in general, which was the intent of doing the examples in the first place.

Any thoughts?

-Dan