Chapter 3.3, Problem 26E

Let $A$ be a given nonempty set. As noted in Example 2 of section 3.1, $S(A)$ is a group with respect to mapping composition. For a fixed element

$a$

In $A$, let $H_a$ denote the set of all $f\in S(A)$ such that $f(a)=a$.Prove that $H_a$ is a subgroup of $S(A)$.

From Example 2 of section 3.1: Set $A$ is a one – to – one mapping from $A$ onto $A$ and $S(A)$ denotes the set of all permutations on $A$.

$S(A)$ is closed with respect to binary operation $\circ$ of mapping composition. The identity mapping $I(A)$ in $S(A)$,

$f\circ I_A=f=I_A\circ f$ for all $f\in S(A)$, and also that each $f\in S(A)$ has an inverse in $S(A)$.

Thus we conclude that $S(A)$ is a group with respect to composition of mapping.