The Cayley table for the group of symmetries of a triangle, D-3 , is given below.

The highlighted entries are special. They form a closed set, with each member of the set being composed with another member of the set to form some other member of the set. We say elements of a group which act in this way form a subgroup.

We writeto indicate that the set is a subgroup of

This is not the only subgroup of

andare all subgroups of

Any set of elements S of a group G must satisfy the following three condictions, called the subgroup axoims, to form a subgroup.

Ifthen

If

In addition, the number of elements in any subgroup must divide the number of elements of G. D-3 has 6 elements. The factors of 6 are 1, 2, 3 and 6, so the only possible subgroups ofmust have 1, 2, 3 or 6 elements. In fact, for every groupwithelements, there is always a subgroup withelements.

The setalways forms a subgroup. It is called the trivial subgroup.

Subgroups of abelian groups are also abelian and subgroups of cyclic groups are always cyclic.