Question

Do the odd permutations of S_n form a subgroup of S_n? If so, provide a proof. If not, provide a counterexample with justification

Answer 1

Definition : A permutation f in Sn is called even if f can be written as a product of an even number of transpositions (a cycle of length 2)

A permutation f in Sn is odd if f can be written as a product of an odd number of transpositions.

Let, x & y be 2 odd permutations in Sn where x is the product of m transpositions & y is the product of p transpositions in Sn, where m & p are odd.

So, the product of these 2 odd permutations in Sn is x•y where x•y consists of m+p transpositions.

Since, m & p are odd, so, m+p is even.

Hence, x•y is the product of even number of transpositions. Hence, even though both x & y are odd permutations, x•y is an even permutation.

So, the set of all odd permutations in Sn is not closed under product in Sn.

Hence, the set of all odd permutations in Sn does not form a subgroup of Sn.