Subgroup

Some authors also exclude the trivial group from being proper (that is, H ≠ {e}​).

If the group operation is instead denoted by addition, then closed under products should be replaced by closed under addition, which is the condition that for every a and b in H, the sum a + b is in H, and closed under inverses should be edited to say that for every a in H, the inverse −a is in H. Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H}.

Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the equivalence relation a1 ~ a2 if and only if ⁠

⁠ is in H. The number of left cosets of H is called the index of H in G and is denoted by [G : H].

Lagrange's theorem states that for a finite group G and a subgroup H, where |G| and |H| denote the orders of G and H, respectively.

More generally, if p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal.

Each group (except those of cardinality 1 and 2) is represented by its Cayley table.