# Homework Help: Group inverses

1. Feb 1, 2009

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

x is in the arbitrary multiplicative group, and a,b are positive integers.
given that
$$x^{a+b} = x^ax^b$$ and $$(x^a)^b =x ^{ab}$$
show
that
$$(x^a)^-1 = x^{-a}$$

2. Relevant equations
na

3. The attempt at a solution
Induction:

I) (x)^{-1} = x ^{-1}
II) Assume $$(x^n)^{-1} = x^{-n}$$, to prove that $$(x^{n+1})^{-1} = x^{-(n+1)}$$.

$$(x^{n+1})^{-1}) = (x^nx)^{-1} = x^{-1}x^{-n} = x^{-n-1}$$

Is the last step justified?

2. Feb 1, 2009

### Hurkyl

Staff Emeritus
Seeing how you didn't provide a justification, no.

More seriously, if you cannot see a rigorous reason why that last step should be true, then you definitely haven't written a valid proof.

3. Feb 1, 2009

"$$x^{-1}x^{-1}x^{-1} \ldots x^{-1} \text{ n terms }$$"
$$x^{n} = x^{-1}x^{-(n-1)}$$