Page 26 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Abelian Group of Commutative Group

A group (G, o) is said to be abelian or commutative if the composition ‘o’ is commutative, i.e., if
a o b = b o a a, b G
A group which is not abelian is called non-abelian.

Examples:
(i) The structures (N, +) and (N, ×) 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 (G ) and (G ) in the former case, and (G ) in the latter case, are not
3 4 4
satisfied.
(ii) The structure (Z, +) 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 O
belongs to Z, and the inverse of every element a, viz., –a belongs to Z. This is known as
additive group of integers.

The structure (Z, ×), i.e., the set of integers with the multiplication composition does not
form a group, as the axiom (G ) is not satisfied.
4
(iii) The structures (Q, +), (R, +), (C, +) 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.
(iv) The structure (Q , x) is a group, where Q is the set of non-zero rational numbers. This is so
0 0
because the operation is associative, the multiplicative identity 1 belongs to Q , and the
0
multiplicative inverse of every element a in the set is 1/a, which also belongs to Q . This
0
is known as the multiplicative group of non-zero rationals.
Obviously (R , X) and (C , X) are groups, where R and C are respectively the sets of non-
0 0 0 0
zero real numbers and non-zero complex numbers.
+
(v) The structure (Q , ×) is a group, where Q is the set of positive rational numbers. It can
+
easily be seen that all the postulates of a group are satisfied.
+
+
Similarly, the structure (R , ×) is a group, where R is the set of positive real numbers.
(vi) The groups in (ii), (iii), (iv) and (v) above are all abelian groups, since addition and
multiplication are both commutative operations in numbers.

Finite and Infinite Groups

If a group contains a finite number of distinct elements, it is called finite group otherwise an
infinite group.

In other words, a group (G, 0) is said to be finite or infinite according as the underlying set G is
finite or infinite.

Order of a Group

The number of elements in a finite group is called the order of the group. An infinite group is
said to be of infinite order.

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