# Definition of Group

An algebraic structure $\left( {G, * } \right)$, where $G$ is a non-empty set with a binary operation “$*$” defined on it is said to be a group; if the binary operation satisfies the following axioms (called group axioms).

(G1) Closure Axiom. $G$ is closed under the operation $*$, i.e., $a * b \in G$, for all $a,b \in G$.

(G2) Associative Axiom. The binary operation $*$ is associative. i.e., $\left( {a * b} \right) * c = a * \left( {b * c} \right)$   $\forall a,b \in G$.

(G3) Identity Axiom. There exists an element $e \in G$ such that $e * a = a * e = a$   $\forall a \in G$. The element $e$ is called the identity of “$*$” in $G$.

(G4) Inverse Axiom. Each element of $G$ possesses inverse, i.e., for each element $a \in G$, there exists an element $b \in G$ such that $b * a = a * b = e$
The element $b$ is then called the inverse of a with respect to “$*$” and we write $b = {a^{ - 1}}$. Thus ${a^{ - 1}}$ is an element of $G$ such that ${a^{ - 1}} * a = a * {a^{ - 1}} = e$

Examples:

1. The structures $\left( {\mathbb{N}, + } \right)$ and $\left( {\mathbb{N}, \times } \right)$ are not groups i.e., the set of natural numbers considered with the addition composition or the multiplication composition, does not form a group. For, the postulate (G3) and (G4) in the former case, and (G4) in the latter case, are not satisfied.
2. The structure $\left( {\mathbb{Z}, + } \right)$ is a group, i.e., the set of integers with the addition composition is a group. This is so because addition in numbers is associative, the additive identity $0$ belongs to $\mathbb{Z}$, and the inverse of every element $a$, viz. $- a$ belongs to $\mathbb{Z}$. This is known as additive group of integers.
The structure $\left( {\mathbb{Z}, \times } \right)$, i.e., the set of integers with the multiplication composition does not form a group, as the axiom (G4) is not satisfied.
3. The structures $\left( {\mathbb{Z}, + } \right),\left( {\mathbb{R}, + } \right),\left( {\mathbb{C}, + } \right)$ are all groups, i.e., the sets of rational numbers, real numbers, complex numbers, each with the additive composition, form a group. But the same sets with the multiplication composition do not form a group, for the multiplicative inverse of the number zero does not exist in any of them.
The structure $\left( {{\mathbb{Q}_o}, \times } \right)$ is a group, where ${\mathbb{Q}_o}$ is the set of non-zero rational numbers. This is so because the operation is associative, the multiplicative identity $1$ belongs to ${\mathbb{Q}_o}$, and the multiplicative inverse of every element $a$ in