GROUPS 5

Finite Groups and Group Tables

Finite group - Group where the set G has finite element.

The rules :

1. List the identity first (use definition G2).
2. To find an inverse elements for each element e, a (use definition G3) a' must be either a or e.
   
3. All axioms are satisfied except possibly the associative property. Checking associativity on a case by case basis.

Note : each element of a group must appear once and only one in each row and column.


Example 1 : The smallest finite set, {e}.

- Group of one element set (satisfied G1, G2, G3)


Solution :

Let e element of A
e*e element of A

G1 : e element of A

(e*e)*e = e*(e*e)
= e*e
= e

G2 :

e*e =e

G3 :

e*e = e


(e*a)*a = a*a = e , e*(a*a) = e*e = e
(e*a)*e = a*e = a , e*(a*e) = e*a = a
(a*e)*a = a*a = e , a*(e*a) = a*a = e
(a*e)*e = a*e = a , a*(e*e) = a*e = a

Thus,

(e*a)*a = e*(a*a) , (a*e)*a = a*(e*a)
(e*a)*e = e*(a*e) , (a*e)*e = a*(e*e)

Hence, {e,a} satisfies the associative property.

Example 2 :

Let a set of two element {e,a}. Construct a table for a binary operation * on {e,a} that gives a group structure on {e,a}.


Soluiton :