Prove that:
a) In a group (ab)^-1=b^-1 * a^-1
b) A group G is Abelian iff (ab)^-1=a^-1 * b^-1
c) In a group (a^-1)^-1=a for all a
Multiply it to the right by (ab)
Since it is a group, * is associative.
So we can write :
So we have
Therefore...
Abelian means that * is commutative.b) A group G is Abelian iff (ab)^-1=a^-1 * b^-1
If it is Abelian, by the previous relation.
If , then since , we have
Thus it is commutative --> Abelian
The equivalence is proved.
c) In a group (a^-1)^-1=a for all a
Sorry for the wording & the method... just trying myself to explain