What is index of a group?

The index of a group is a measure of the size of the group relative to the size of its subgroups. Specifically, the index of a subgroup H with respect to a group G is the ratio of the number of elements in G that are not in H to the number of elements in H. This ratio is denoted by |G:H| and can be calculated as |G:H| = |G|/|H|, where |G| and |H| are the orders (or sizes) of G and H, respectively.

For example, if G is a group of order 12 and H is a subgroup of order 4, then |G:H| = |G|/|H| = 12/4 = 3. This means that there are 3 cosets of H in G, each of which has order 4.

The index of a subgroup can give us information about the structure of the group. For instance, if |G:H| is prime, then H must be a normal subgroup of G. This is a consequence of the fact that the cosets of a normal subgroup form a partition of the group, and the number of cosets is equal to the index.

Moreover, a famous result in group theory known as the Lagrange's theorem states that the order of any subgroup of a group must divide the order of the group. This can help us determine if a given subgroup is possible or not.