Subgroups
The Cayley table for the group of symmetries of a triangle, D-3 , is given below.
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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 write
to indicate that the set is a subgroup of
This is not the only subgroup of
and
are 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.
If
then
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 of
must have 1, 2, 3 or 6 elements. In fact, for every group
with
elements, there is always a subgroup
with
elements.
The set
always forms a subgroup. It is called the trivial subgroup.
Subgroups of abelian groups are also abelian and subgroups of cyclic groups are always cyclic.
