Algebra - Group Theory Q&A | TN TRB PG | CSIR NET | SET.
Group
Definition :
A non-empty set G
is said to form a group if in G there is
defined a binary operation, called the product and denoted by (·)
such that
1.
a,
b ∈ G ⇒ a · b
∈ G (Closure axiom)
2.
a,
b, c ∈ G ⇒ a · (b
· c) = (a · b) · c (Associative axiom)
3.
There
exists an element e ∈ G such that a · e = e · a = a,
∀ a
∈ G (Existence of identity)
4.
∀ a
∈ G there exists an element a−1 ∈ G such that a·a−1 =
a−1 ·a = e (Existence of inverse).
(Z,
+) is an infinite abelian group.
Also
(Q, +), (R , +), (C, +) are infinite abelian groups.