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.


Proof :

(To show the identity element in G is unique)

Let e and f be identity of group (G,*)
(T0 show e = f)

Since e is the identity, then

e*f = f*e = f ---1

Now, since f also is identity, the

e*f = f*e = e ---2

Hence, by equations 1 and 2

f = e

(To show the inverse element is unique in (G,*))

Let a' and a" are inverse of a element of G.

(To show a' = a")

Since a' is an inverse of a in G, then

a'*a = a*a' = e ---3

Since a" is also an inverse of a in G, then

a"*a = a*a" = e ---4

Hence, by equation 3 and 4

a'*a = a"*a
a' = a"
(by right cancellation law)

Then, the inverse of a in group (G,*) is unique.


Corollary 1

Let G be a group. For all a,b element of G, we have

(a*b)' = b'*a'.

Notes :

Some binary structure with weaker axioms :

• Semigroup - A set with an associative binary structure.
• Monoid - A semigroup that has an identity for binary operation.

Every group both semigroup and monoid!


Example :

Let (G,*) be a group and a element of G. Prove that,

if a*a = a then a = e.

Solution :

a*a = a then a = e

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

sources : http://www.gpc.edu/~duniss/images/PE07231_.gif