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.

GROUPS 6

Example 3 :

The following examples give groups with their respective tables.

1. Group V = {e,a,b,c} under multiplication (.). This group is the Klein 4-group. (notation V comes from the German word for four).

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.

GROUPS 4

Theorem 3 ( The uniqueness of identity & inverse elements in G )

In a group G with binary operation *, there is only one element e in G such that

e*x = x*e = x

for all x in G.

Likewise, for each a in G, there is only one element a' in G that

a'*a = a*a' = e

In summary, the identity & inverse of each element are unique in a group.

GROUPS 3

Elementary Properties of Groups


Theorem 1 (Left & Right Cancellation Laws)

If G is a group with binary operation *, then the left and right cancellation laws hold in G, that is

a*b = a*c implies b = c, and
b*a = c*a implies b = c for all a,b,c in G

Proof:

(To prove the left cancellation law)
To show if a*b = a*c then b = c
Let a*b = a*c

Since G is a group, G3 is satisfied. Then, for all a' in G such that,

a'*(a*b) = a'*(a*c) --G3
(a'*a)*b = (a'*a)*c --G1,* associative
e*b = e*c --by def. of a' in G3
b = c

Similarly, we can prove the right cancellation law.