GROUPS 7

Example 5 :

Complete the following table so that * is commutative binary operation on the set S = {a,b,c,d}.


From the table, there is no identity and inverse element.(G1 and G2 are not satisfied).
Therefore, it is not a group.



Example 6 :

Let S = {a,b,c}. Define the binary operation * on S by the following table.


Determine
(a) the identity for *,
(b) the inverse of a,(c) the inverse of b.


Solution :

(a) Column c and row have the same elements as the header of the table. So, c is the identity for set S.
(b) From the table, a*b = c = b*a. So, the inverse of a is b.
(c) Also, the inverse of b must be a.


Order of a Group and Order of an Element

Definition 3 (Order of a group)
Let G be a finite group. The order of group G, defined by |G|, is the number of elements in G.

Definition 4 (Order of an element)
Let G be a group and a element of G. If there exists a positive integer n such that

then, the smallest such positive integer is called an order of a.

• The order of an element a element of G is denoted by o(a).
• If e is the identity of group G, then o(e) = 1
• Conversely, if an element of G of order 1 then, the element must be the identity of G.

sources : http://www.msmichelle.org/images/look_feel/math_graphic.jpg