Page 30 - DMTH403_ABSTRACT_ALGEBRA

Unit 2: Groups

Similarly, an element may not have an inverse with respect to a binary operation. For example, Notes
2 " Z has no inverse with respect to multiplication on Z, does it?
Now let us consider the following examples.

Example: If the binary operation # : R × R  R is defined by a  b = a + b – 1, prove that
 has an identity. If x  R, determine the inverse of x with respect to , if it exists.

Solution: We are looking for some e  R such that a  e = a = e  a  a  R. So we want e  R such
that a + e – 1 = a  a  R. Obviously, e = 1 will satisfy this. Also, 1  a= a  a  R. Hence, 1 is the
identity element of .
For x  R, if b is the inverse of x, we should have b  x = 1.
i.e., b + x – 1 = 1, i.e., b = 2 – x. Indeed, (2 – x)  x = (2 – x) + x – 1 = l.

Also, x  (2 – x) = x + 2 – x – 1 =l. So, x = 2 – x.
–1

Example: Let S be a non-empty set. Consider  (S), the set of all subsets of S. Are  and
 commutative or associative operations on (S)? Do identity elements and inverses of elements
of (S) exist with respect to these operations?
Solution: Since A  B = B  A and A  B = B  A  A, B  (S), the operations of union and
intersection are commutative. You can see that the empty set  and the set S are the identities of
the operations of union and intersection, respectively. Since S  , there is no B   (S) such that
S  B = . In fact, for any A  (S) such that A # IS, A does not have an inverse with respect to
union. Similarly, any proper subset of S does not have an inverse with respect to intersection.

When the set S under consideration is small, we can represent the way a binary operation on S
acts by a table.

2.1.1 Operation ’.’ Table

Let S be a finite set and * be a binary operation on S. We can represent the binary operation by
a square table, called an operation table or a Cayley table. The Cayley table is named after the
famous mathematician Arthur Cayley (1821-1895).

To write this table, we first list the elements of S vertically as well as horizontally, in the same
order. Then we write a * b in the table at the intersection of the row headed by a and the column
headed by b.

For example, if S = (–1, 0, 1) and the binary operation is multiplication, denoted by., then it can
be represented by the following table.

. –1 0 1
–1 (–1) . (–1) (–1) . 0 (–1) . 1
= 1 = 0 = –1
0 0 . (–1) 0.0 0.1
= 0 = 0 = –1
1 1 . (–1) 1.0 1.1
= –1 = 0 =1

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