Power of Product with Inverse

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Theorem

Let $G$ be a group whose identity is $e$.

Let $a, b \in G: a b = b a^{-1}$.


Then:

$\forall n \in \Z: a^n b = b a^{-n}$


Proof

Proof by induction:


For all $n \in \Z$, let $P \left({n}\right)$ be the proposition $a^n b = b a^{-n}$.

$P(0)$ is trivially true, as $a^0 b = e b = b = b e = b a^{-0}$.


Basis for the Induction

$P(1)$ is true, as this is the given relation between $a$ and $b$:

$a b = b a^{-1}$

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:

$a^k b = b a^{-k}$


Then we need to show:

$a^{k+1} b = b a^{-\left({k+1}\right)}$


Induction Step

This is our induction step:

\(\displaystyle a^{k+1} b\) \(=\) \(\displaystyle a \left({a^k b}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle a \left({b a^{-k} }\right)\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \left({a b}\right) a^{-k}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({b a^{-1} }\right) a^{-k}\) Basis for the Induction
\(\displaystyle \) \(=\) \(\displaystyle b \left({a^{-1} a^{-k} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle b a^{-\left({k+1}\right)}\)


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


Therefore $\forall n \in \N: a^n b = b a^{-n}$.


Now we show that $P \left({-1}\right)$ holds, i.e. that $a^{-1} b = b a$.

\(\displaystyle a b\) \(=\) \(\displaystyle b a^{-1}\)
\(\displaystyle \iff \ \ \) \(\displaystyle a b a\) \(=\) \(\displaystyle b\)
\(\displaystyle \iff \ \ \) \(\displaystyle b a\) \(=\) \(\displaystyle a^{-1} b\)

... thus showing that $P \left({-1}\right)$ holds.


The proof that $P \left({n}\right)$ holds for all $n \in \Z: n < 0$ then follows by induction, similarly to the proof for $n > 0$.

$\blacksquare$