Binary Operations

A mapping \[ *: S \times S \rightarrow S \] is called a binary operation on the set \(S\).

Thus, a binary operation on \(S\) assigns exactly one element of \(S\) to every ordered pair of elements of \(S\). If \(a,b\in S\), then the element assigned to \((a,b)\) is usually denoted by \(a*b\).

Binary operations are generally represented by symbols such as \( * \), \( \cdot \), \(+\), and \(\circ\), rather than by letters such as \(f\) and \(g\).

Semigroup

A semigroup is a nonempty set \(S\), together with an associative binary operation on \(S\). Thus, for all \(a,b,c\in S\),

\[a*(b*c)=(a*b)*c.\]

Example: Each of the sets \(\mathbb{Z}\), \(\mathbb{R}\), and \(\mathbb{C}\), equipped with the usual operation of addition or multiplication, forms a semigroup.

Identity Element

Let \((S,\cdot)\) be a semigroup. An element \(e\in S\) is called an identity element of the semigroup if

\[e\cdot x=x=x\cdot e \qquad \text{for every }x\in S.\]

Inverse of an Element

Let \((S,\cdot)\) be a semigroup with identity element \(e\), and let \(a\in S\). If there exists an element \(b\in S\) such that

\[a\cdot b=e=b\cdot a,\]

then \(b\) is called the inverse of \(a\). It is denoted by \(a^{-1}\).

Group

A nonempty set \(G\), together with a binary operation \(\cdot\) on \(G\), is called a group if the following axioms are satisfied:

• Closure: For all \(a,b\in G\), we have \(a\cdot b\in G\).
• Associativity: For all \(a,b,c\in G\), \[a\cdot(b\cdot c)=(a\cdot b)\cdot c.\]
• Existence of an identity: There exists an element \(e\in G\) such that \[e\cdot a=a=a\cdot e\qquad\text{for every }a\in G.\]
• Existence of inverses: For every \(a\in G\), there exists an element \(a^{-1}\in G\) such that \[a\cdot a^{-1}=e=a^{-1}\cdot a.\]

Abelian Group

A group \((G,\cdot)\) is called an abelian group or a commutative group if

\[a\cdot b=b\cdot a\qquad\text{for all }a,b\in G.\]

Examples of Groups

• The sets \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\), and \(\mathbb{C}\) form abelian groups under the usual operation of addition.
• The sets \(\mathbb{Q}^{*}\), \(\mathbb{R}^{*}\), and \(\mathbb{C}^{*}\), where the asterisk denotes the set of nonzero elements, form abelian groups under multiplication.
• The set \(\mathbb{Z}^{*}\) of nonzero integers does not form a group under multiplication because most nonzero integers do not have multiplicative inverses in \(\mathbb{Z}\). However, the set of units \(\{-1,1\}\) forms a group under multiplication.
• The set of integers modulo \(n\), denoted by \(\mathbb{Z}_n\), forms an abelian group under addition modulo \(n\).
• The set of all permutations of a nonempty set forms a group under composition of functions.
• The set of all invertible \(n\times n\) matrices over a field \(F\), denoted by \(GL(n,F)\), forms a group under matrix multiplication. It is called the general linear group of degree \(n\) over \(F\).