Properties of a Group with Proofs

In this class, We discuss Properties of a Group with Proofs.

The reader should have prior knowledge of group. Click Here.

Properties of a Group:

1) The identity element of a group is unique.

Assume e1, e2 be two identity elements.

a * e1 = e1 * a = a

e2 * e1 = e2 = e1 * e2

The above statement happens only if e1 = e2.

Hence one identity element exists.

2) Every element in a group G has unique inverse.

Assume a1 and a2 are two inverse for a.

a * a1 = e

a * a2 = e

a1 = a1 * e

a1 = a1 * (a * a2) placing the e value from above.

a1 = (a1 * a)* a2

a1 = e * a2

a1 = a2

3) In a group (a’)’ = a

We know a * a’ = a’ * a = e

Take a’ * a = e

Both sidess use (a’)’

we get (a’)’ * (a’ * a) = (a’)’ * e

((a’)’ * a’) * a = (a’)’

e * a = (a’)’

a = (a’)’

4) In a group (a * b)’ = b’ * a’

a * a’ = a’ * a = e

b * b’ = b’ * b = e

Assume (a * b)’ = b’ * a’

(a * b) * (b’ * a’) = e

(a *(b * b’) * a’)

a * e * a’

a * a’

= e

Our assumption is true.

5) Cancellation law:

a * b = a* c -> b = c

b * a = c * a -> b = c

Solution:

a* b = a* c

a’ * (a* b) = a’ * (a* c)

(a’ * a)* b = (a’ * a)* c

e * b = e * c

b = c