Page 22 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 1.2 Groups

The theory of groups, an important part in present day mathematics, started early in nineteenth
century in connection with the solutions of algebraic equations. Originally a group was the set
of all permutations of the roots of an algebraic equation which has the property that combination
of any two of these permutations again belongs to the set. Later the idea was generalized to the
concept of an abstract group. An abstract group is essentially the study of a set with an operation
defined on it. Group theory has many useful applications both within and outside mathematics.
Group arise in a number of apparently unconnected subjects. In fact they appear in crystallography
and quantum mechanics, in geometry and topology, in analysis and algebra and even in biology.
Before we start talking of a group it will be fruitful to discuss the binary operation on a set
because these are sets on whose elements algebraic operations can be made. We can obtain a
third element of the set by combining two elements of a set. It is not true always. That is why this
concept needs attention.
Binary Operation on a Set

The concept of binary operation on a set is a generalization of the standard operations like
addition and multiplication on the set of numbers. For instance we know that the operation of
addition (+) gives for any two natural numbers m, n another natural number m + n, similarly the
multiplication operation gives for the pair m, n the number m.n in N again. These types of
operations are found to exist in many other sets. Thus we give the following definition.
Definition

A binary operation to be denoted by ‘o’ on a non-empty set G is a rule which associates to each
pair of elements a, b in G a unique element a o b of G.
Alternatively a binary operation ‘o’ on G is a mapping from G × G to G i.e. o : G × G G where
the image of (a, b) of G × G under ‘o’, i.e., o (a, b), is denoted by a o b.
Thus in simple language we may say that a binary operation on a set tells us how to combine any
two elements of the set to get a unique element, again of the same set.

If an operation ‘o’ is binary on a set G, we say that G is closed or closure property is satisfied in G,
with respect to the operation ‘o’.

Examples:
(i) Usual addition (+) is binary operation on N, because if m, n N then m + n N as we know
that sum of two natural numbers is again a natural number. But the usual substraction
(–) is not binary operation on N because if m, n N then m – n may not belongs to N. For
example if m = 5 and n = 6 their m – n = 5 – 6 = –1 which does not belong to N.
(ii) Usual addition (+) and usual substraction (–) both are binary operations on Z because if
m, n Z then m + n Z and m – n Z.

(iii) Union, intersection and difference are binary operations on P(A), the power set of A.
(iv) Vector product is a binary operation on the set of all 3-dimensional Vectors but the dot
product is not a binary operation as the dot product is not a vector but a scalar.

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