Definition#

A group $(G, * )$ consists of a set $G$ and a binary operation $*$ such that

• $G$ is closed under $*$
  This means that if $g * h$ belongs to $G$ for all $g, h \in G$.
• Associativity
  This implies that $g_1 * (g_2 * g_3) = (g_1 * g_2) * g_3$ for all $g_1, g_2, g_3 \in G$.
• There exists an identity element
  There exists an element $e$ such that $g * e = e * g = g$ for all $g \in G$.
• Every element has an inverse.
  For every element $g \in G$, there exists an element $h$ such that $g * h = h * g = e$.

Some examples of groups are $(\mathbb{Z}, +)$, $(\mathbb{Z}_0, \times)$.

If the group operation is commutative, then we say the group is abelian.

The order of the group is the number of elements in the group. $|G|$.

The order of an element $g$ is the minimum integer $n$ such that $g^n = e$.