Prove that every element of order 2 in is the square of an element of order 4 in .
We know that every element of order 2 in (hence also ) is a product of commuting 2-cycles by a previous theorem. Write of order 2 as . Note that the number of 2-cycles in the decomposition of is even since .
Note that . Thus using a previous theorem, and the element has order 4 in by §1.3 #15.