Subgroups of the symmetric group?

Question

The symmetric group Sn consists of the n! permutations of n distinct objects under the binary operation of combining permutations. Each permutation is either even or odd. Prove that in every subgroup of Sn, each element is even or exactly half the elements are even.

Answer 1

Suppose there's at least one odd permutation p.

Look at the mapping from S-->S that sends any s -->ps

Odd permutations are mapped to even ones and vice versa.

So there's a 1-1 correspondence between odd and even permutations. So the number of each is the same.

Q.E.D.

Answer 2

you're taking the attitude of the dihedral and study it to the subgroups.The S5 must be inverted to supply you the reflect photograph of the isomorphic subgroups of 8,10 and 12.Then take the damaging log of each and every attitude to evaluation with the different subgroups.this might enable you to make certain to relationship of the S5 team that are non isomorphic to the propose deviation of each and every subset in terms of diverse non symetric dihedrals. See, incredibly undemanding.