Section 8.6
Small groups
In this section, we determine all groups
of order less than 12.
First we introduce some examples.
Definition
- The dihedral group of order 2n is the group
Dn = {ciaj
| i = 0, 1, ..., n - 1, j = 0, 1}
with multiplication determined by cn = a2 = 1
and
ac = cn-1a.
- The quaternion group is the group of order 8 consisting of the following invertible
quaternions.
Q = {1, -1, i, -i, j, -j, k, -k}.
Next we collect some useful results.
Let G be a group.
Lemma
- If every element of G\{e} has order 2, then G is commutative.
- If G is finite and p is a prime number dividing |G|,
then G has an element of order p.