G is the group , the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to . There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, non-overlapping sets. Thus the index [G : H] is 4.
In the mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite groupG, then |H| is a divisor of |G|, i.e. the order (number of elements) of every subgroup H divides the order of group G.
The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite group , not only is an integer, but its value is the index, defined as the number of left cosets of in .
Lagrange's theorem — If H is a subgroup of a group G, then
This variant holds even if is infinite, provided that , , and are interpreted as cardinal numbers.