Let A be an infinite set. Let H be the set of all S A

Let A be an infinite set. Let H be the set of all σ ∈ S_A such that the number of elements moved by a (see Exercise 30) is finite. Show that H is a subgroup of S_n.

Data from exercise 30

Let σ be a permutation of a set A. We shall say "σ moves a ∈ A" if σ(a) ≠ a. If A is a finite set, how many elements are moved by a cycle σ ∈ S_A of length n?