Let A be an infinite set. Let K be the set of all S A that

Let A be an infinite set. Let K be the set of all σ ∈S_A that move (see Exercise 30) at most 50 elements of A. Is K a subgroup of S_A? Why?

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?