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Prove. In a group a left inverse is also a right inverse.
Group postulates
G1 Closure
G2 Associative law (ab)c = a(bc)
G3 Existence of a left identity ea = a for every a
G4 Existence of a left inverse for each element a a-1a = e
Proof. From postulate G4,
1) a-1a = e
Right multiplying by a-1 gives
2) a-1aa-1 = ea-1
From G3, ea-1 = a-1, 2) becomes
3) a-1aa-1 = a-1
By postulate G4 the inverse element a-1 has some left inverse although we don’t know what it is. Denote it by a*. Left-multiply 3) by a*
4) a*a-1aa-1 = a*a-1
Since a* is a left inverse of a-1, a*a-1 = e, so 4) becomes
5) eaa-1 = e
or
6) aa-1 = e
Thus, from 1) and 6)
7) e = a-1a = aa-1
and every left inverse is also a right inverse.
Prove. In a group, a left identity is also a right identity.
Since e = a-1a
8) ae = a a-1a
From 6) above, e = aa-1. Substituting e for aa-1 in 8) we get
9) ae = ea = a
and thus the left identity is also a right identity.
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