In: Advanced Math
How many elements of order 2 are there in S5 and in S6? How many elements of order 2 are there in Sn? (abstract algebra)
Elements of order 2 in S5 could consist of 1 two-cycles or the product of 2 two-cycles. There are, 5C2 = 10 ways to create a 2-cycle. Then there are 3C2 = 3 ways to create a second 2-cycle. So, there are 10 single 2-cycles, there are 10.3/2 = 15 pairs of disjoint 2-cycles (divide by 2 since either 2-cycle could be listed first).
So, there are 10 + 15 = 25 elements of order 2 in S5.
Elements of order 2 in S6 could consist of 1, 2, or 3
two-cycles. There are 6C2 = 15 ways to create
a 2-cycle. Then there are 4C2 = 6 ways to
create a second 2-cycle. Only a single way remains to create
a third 2-cycle. So there are 15 single 2-cycles, there are 15 ·
6/2 = 45 pairs of disjoint 2-cycles (divide by 2 since either
2-cycle could be listed first), and 15 · 6/6 = 15 triples of
disjoint 2-cycles (3! = 6 ways of ordering 3 items).
Thus there are 15 + 45 + 15 = 75 elements of order 2 in
S6.