# Properties of Permutations of a Set A

Let $A$ be a finite set, and $B$ a subset of $A$. Let $G$ be the subset of $S_A$ consisting of all the permutations $f$ of $A$ such that $f(x)\in B$ for every $x\in B$. Prove that $G$ is a subgroup of $S_A$.

Since $A$ is a finite set we only have to prove that $G$ is closed under the operation. But I am unsure of how to go about showing that. I know that I have to show that $g_1\circ g_2\in G$ for $g_1,g_2\in G$.

• If $x\in B$, then $g_2(x)\in B$, and hence $g_1(g_2(x)) \in B$ – Prahlad Vaidyanathan Oct 28 '13 at 18:01
• Okay thanks for wording that. I knew it was along those lines but I couldn't figure out to word it correctly. :) – Student Oct 28 '13 at 18:04

If $x\in B$, then $g_2(x)\in B$, and hence $g_1(g_2(x))\in B$ – Prahlad Vaidyanathan Oct 28 at 18:01
To beef up this "answer", I'll add that the statement may fail for infinite sets. For example, if $A=\mathbb Z$ and $B$ is the set of positive integers, then $g(x)=x+1$ has the property $x\in B\implies g(x)\in B$ but its inverse does not.