Thread: Prove

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

Originally Posted by mandy123

Prove that:
a) In a group (ab)^-1=b^-1 * a^-1

Multiply it to the right by (ab)



Since it is a group, * is associative.
So we can write :




So we have
Therefore...

b) A group G is Abelian iff (ab)^-1=a^-1 * b^-1

Abelian means that * is commutative.

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

Thank you so much!!! This helps me a lot!!
I like your animated cow by the way!!!

Originally Posted by mandy123

c) In a group (a^-1)^-1=a for all a

therefore inverse of is .
Thus, .