Index of a Subgroup Let a subgroup H of a group G be non-empty. The index of H (written as Index(H)) is the number of cosets of H in G. In simpler terms,...
Let a subgroup H of a group G be non-empty. The index of H (written as Index(H)) is the number of cosets of H in G. In simpler terms, it tells you how many different "pieces" you can pick from the group G to form subsets where each element belongs to the same coset.
Formally:
Index(H) = {g ∈ G | g ∈ coset(h) for every h ∈ H}
where:
coset(h) is the set of elements in G that are left-multiplied by h.
coset(h) is a subset of G.
Examples:
Index(H) = 2 if H = {e, a} is the subgroup of G = Z_4 consisting of the even elements. In this case, there are two cosets: the coset of e and the coset of a.
Index(H) = 3 if H = {e, b, c} is a subgroup of G = Z_6. In this case, there are three cosets: one for each element in H.
Index(H) = 0 if H = {e} is the subgroup of G = Z_4 consisting only of the identity element.
Index(H) = 1 if H = {e, a, b} is a subgroup of G = Z_6. In this case, there is only one coset, which contains all the elements of H.
Significance:
The index of a subgroup is an important measure of the "complexity" of that subgroup. It tells you how many "independent" elements there are within the group, which can be helpful for understanding the structure of the group and its elements. Additionally, knowing the index can help determine if a subgroup is normal or abelian