# Homework Help: Prove (a^(-1))^(-1) = a

1. Dec 17, 2012

### Zondrina

1. The problem statement, all variables and given/known data

Trying to run through every problem i can in my book in preparation for my exam. I've solved this one before, but it slipped my mind how to do it :

2. Relevant equations

Working in a group, so group axioms I suppose.

3. The attempt at a solution

I forgot where to start this one off, I was thinking :

e = aa-1
a-1e = a-1aa-1

That wont get me anywhere though, any pointers would be appreciated.

2. Dec 17, 2012

### micromass

Do you know that every element in a group has a unique inverse?? You can use this by showing that $a$ and $(a^{-1})^{-1}$ are inverses of the same element. So they must be equal.

3. Dec 17, 2012

### Zondrina

Ah, so what you're saying is if a is a group element, then it has an inverse which is also a group element denoted by a-1.

Since a-1 is also a group element and the inverse of a, then (a-1)-1 is also a group element and is the inverse of a-1.

4. Dec 17, 2012

### micromass

Yes. So $a^{-1}$ has two inverses. Those inverses must equal.

5. Dec 17, 2012

### I like Serena

Hi Zondrina!

In a group every element has to have an inverse.
Now suppose we have b=a-1.
Then according to the group axioms we have: ab=e
What happens if you multiply the left and right hand sides with b-1?