# Thread: Exponent Laws for Groups

1. ## Exponent Laws for Groups

How do you prove the following exponent laws for groups?

let x be an element of group G. Then for all integers m and n:

(i) x^-n = (x^-1)^n
(ii) x^m . x^n = x^(m+n)
(iii) x^m . x^n = x^n . x^m
(iv) (x^m)^n = x^(mn)

They seem self-evident, but I'm not quite sure how to prove them, especially when m and n are negative integers.

2. You need to write down the definition. For $\displaystyle n\geq 1$, the expression $\displaystyle x^n$ is defined to be $\displaystyle x\cdot x\cdots x$ ($\displaystyle n$ times). We also define $\displaystyle x^0=e$ and $\displaystyle x^{-n}$ to be the inverse of $\displaystyle x^n$.

Try using these to prove your statements.

3. I think I can prove statements (ii), (iii) and (iv) for all non-negative integers m and n using induction on either m or n. However, I don't know how to prove the statements for negative integers.

4. Originally Posted by Alfie
How do you prove the following exponent laws for groups?

let x be an element of group G. Then for all integers m and n:

(i) x^-n = (x^-1)^n
(ii) x^m . x^n = x^(m+n)
(iii) x^m . x^n = x^n . x^m
(iv) (x^m)^n = x^(mn)

They seem self-evident, but I'm not quite sure how to prove them, especially when m and n are negative integers.
Personally, I would show that $\displaystyle \langle x\rangle\cong \mathbb{Z}/n\mathbb{Z}$ for some $\displaystyle n \in \mathbb{Z}$, or $\displaystyle \langle x\rangle \cong \mathbb{Z}$ in a natural' way. You can then use this isomorphism to prove your result (you need to work out what I mean by natural' (that is, find the isomorphism), because this is really the thing you use; you can't use the fact that the are isomorphic, you use the actual isomorphism).