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 an 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:
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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.
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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 the 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. -
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.