The index of a subgroup of a group is the number of distinct left or right cosets of that subgroup. This is denoted
If and are finite, Lagrange’s theorem states that Hence, in the finite case, the index of in is the order of divided by the order of . Be careful though: and are not always finite. In the infinite case, the index is taken to be the number of cosets of in (this can be finite even if and are infinite).
Suppose is a subgroup of of index 2. Let .
If then and , so
If then is the disjoint union of the left cosets and and the disjoint union of the right cosets and Hence, again
Since for all is normal in The beauty of this result is that it doesn’t assume that or is finite, only that the index of is finite.