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.


Theorem 2 (Unique Solution)

If G is a group with binary operation *, and if a & b are any element of G, then the linear equations

a*x = b and y*a =b

have unique solution x and y in G.

Proof :

Let a,b are any element in G

(To show the existence of at least one solution of a*x = b)

Suppose (a'*b) is one of the solution.
Then,

a*(a'*b)
= (a*a')*b --by G1
= e*b --by def. of G3
= b -- by def. of G2

Thus, (a'*b) is a solution of a*x = b


(Next, to show y = b*a' is a solution of y*a = b)

a*(b*a')
= a*(a'*b) --* is commutative in G
= (a*a')*b -- by G1
= e* b -- by def. of G3= b -- by def. of G2

Thus, (b*a') is a solution of y*a = b


(To show the solution are unique)

Let x' & x" are solution of a'*x = b

Then,

a*x' = b
a*x" = b

Hence,

a*x' = a*x"
x' = x"
(by left cancellation law)

Finally, the uniqueness of y follows similarly.

sources : http://www.maranausd.org/images/pages/N3555/math1.gif