Powers of Elements in Group Direct Product

Theorem
Let $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ be group whose identities are $e_G$ and $e_H$.

Let $\left({G \times H, \circ}\right)$ be the group direct product (either external or internal) of $G$ and $H$.

Then:
 * $\forall n \in \Z: \forall g \in G, h \in H: \left({g, h}\right)^n = \left({g^n, h^n}\right)$

Proof
Proof by induction:

For all $n \in \N$, let $P \left({n}\right)$ be the proposition $\forall g \in G, h \in H: \left({g, h}\right)^n = \left({g^n, h^n}\right)$.


 * $P(0)$ is true, as this just says $\left({g, h}\right)^0 = \left({e_G, e_H}\right)$.

Basis for the Induction

 * $P(1)$ is trivially true, as this just says $\left({g, h}\right) = \left({g, h}\right)$. This is our basis for the induction.

Induction Hypothesis

 * Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:


 * $\left({g, h}\right)^k = \left({g^k, h^k}\right)$

Then we need to show:


 * $\left({g, h}\right)^{k+1} = \left({g^{k+1}, h^{k+1}}\right)$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \N: \forall g \in G, h \in H: \left({g, h}\right)^n = \left({g^n, h^n}\right)$


 * So we have shown the result holds true for all $n \ge 0$.

The result for $n < 0$ follows directly from Powers of Group Elements for negative indices.