### Index of subgroup and coset of subgroup

Explain what the index of a subgroup and a coset of a group are. Also, prove that if N is a subgroup of a group G such that [G: N] = 2, and if "a" and "b" are elements of G, then the product "ab" is an element of N if and only if either (1) both "a" and "b" are elements of N or (2) neither "a" nor "b" is an element of N.