A group consists of a set and a binary operation on that set that meets certain conditions. Groups are a kind of algebraic structure. An algebraic structure consists of one or more sets and operations on those sets. The integers with addition and multiplication are another kind of algebraic structure. This structure consists of a set with two binary operations and is called a Ring. In this course, we study only groups.
The video in Figure 14.1.1 explains why we define a group and gives the definition. After the video, we present the formal definition of a group and give examples.
A pair \((G,\bullet)\) consisting of a set \(G\) and a binary operation \(\bullet:G\times G\to G\) is a commutative group if the following properties hold:
Identity: There is an element \(e\in G\) such that for all \(a\in G\) we have \(a\bullet e=e\bullet a=a\text{.}\) The element \(e\) is called the identity of \((G,\bullet)\text{.}\)
Inverses: For each \(a\in G\) there is \(b\in G\) such that \(a\bullet b = b \bullet a= e\text{,}\) where \(e\) is the identity element in \(G\) with respect to \(\bullet\text{.}\) The element \(b\) is called an inverse of \(a\text{.}\)
Associativity: The operation \(\bullet\) is associative. So, \(a\bullet(b\bullet c) = (a\bullet b)\bullet c\) for all \(a\in G\text{,}\)\(b\in G\text{,}\) and \(c\in G\text{.}\)
Commutative groups are also called abelian groups, after the Norwegian mathematician Niels Abel (1802 — 1829). A group that does not satisfy Item 4 is called a group, or more specifically, a non-commutative group or non-abelian group. In this course, we consider only commutative groups. So when we say group, we mean a commutative group. We call the operation \(\bullet\) of a group \((G,\bullet)\) the group operation of the group.
In the same way, we can show that each element of a group has exactly one inverse for the group operation \(\bullet\text{.}\) So we can speak of the inverse of an element. Recall that we denote the inverse of an element \(a\) for the operation \(\bullet\) by \(a^{-1\bullet}\text{.}\)
Let \((G,\bullet)\) be a group with identity element \(e \in G\text{.}\) Then, for each element \(a \in G\text{,}\) there is exactly one element \(b \in G\) such that \(a\bullet b=e\) and \(b\bullet a=e\text{,}\) implying that the inverse of each element \(a \in G\) is the unique element \(b=a^{-1\bullet}\text{.}\)
Let \((G,\bullet)\) be a group with identity element \(e \in G\text{.}\) Suppose that \(b\in G\) and \(c\in G\) are both inverses of the element \(a\) in \((G,\bullet)\text{.}\) Then:
\begin{align*}
{2} b \amp = b \bullet e \qquad\qquad\amp \amp \text{ since \(e\) is the identity element of \((G,\bullet)\) }\\
\amp = b \bullet (a \bullet c) \qquad\qquad\amp \amp \text{ since \(a\) and \(c\) are inverses in \((G,\bullet)\) }\\
\amp = (b \bullet a) \bullet c \qquad\qquad\amp \amp \text{ since \((G,\bullet)\) is associative }\\
\amp = e \bullet c \qquad\qquad\amp \amp \text{ since \(a\) and \(b\) are inverses in \((G,\bullet)\) }\\
\amp = c \qquad\qquad\amp \amp \text{ since \(e\) is the identity element of \((G,\bullet)\) }
\end{align*}
