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

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- Sep 17th 2008, 06:09 AMmandy123Prove
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 - Sep 17th 2008, 06:17 AMMoo

Multiply it to the right by (ab)

Since it is a group, * is associative.

So we can write :

So we have

Therefore...

Quote:

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.

Quote:

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

Sorry for the wording & the method... just trying myself to explain (Worried) - Sep 17th 2008, 06:22 AMmandy123thanks
Thank you so much!!! This helps me a lot!!

I like your animated cow by the way!!! - Sep 17th 2008, 12:57 PMThePerfectHacker