How many elements of order 2 are there in S5 and in S6? How many elements...

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)

Expert Solution

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.

Colby Messinger answered 2 years ago

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