The following is an exercise question from "Guide to Abstract Algebra, 1st ed, Carol Whitehead".
Decide, giving reasons, whether each of the following subsets of $S_n$ ($n \geq 2$), is or is not a subgroup:
(a) $ \{ \alpha \in S_n : \alpha(1) = 1 \} $
(b) $ \{ \alpha \in S_n : \alpha(1) = 2 \} $
Question: I don't understand the definitions of the subsets.
My thoughts:
Let's consider $S_3$, which contains permutations which map $[1,2,3]$ to $[1,3,2]$ and $[2, 3, 1]$, for example. These can be represented using cycle notation as $(2\;3)$ and $(1\;2\;3)$.
Let's now consider the subset $H = \{ \alpha \in S_n : \alpha(1) = 1 \} $.
Is this all the permutations which preserve the mapping $1 \mapsto 1$ ?
So $(2\;3) \in H$ because it doesn't affect 1, but $(1\;2\;3) \notin H$ because it maps $1 \mapsto 2$.
Is this a correct understanding of the subsets defined in the exercise?