# Inverse of Product/Monoid

## Theorem

Let $\struct {S, \circ}$ be a monoid whose identity is $e$.

Let $a, b \in S$ be invertible for $\circ$, with inverses $a^{-1}, b^{-1}$.

Then $a \circ b$ is invertible for $\circ$, and:

$\paren {a \circ b}^{-1} = b^{-1} \circ a^{-1}$

### General Result

Let $\left({S, \circ}\right)$ be a monoid whose identity is $e$.

Let $a_1, a_2, \ldots, a_n \in S$ be invertible for $\circ$, with inverses $a_1^{-1}, a_2^{-1}, \ldots, a_n^{-1}$.

Then $a_1 \circ a_2 \circ \cdots \circ a_n$ is invertible for $\circ$, and:

$\forall n \in \N_{> 0}: \left({a_1 \circ a_2 \circ \cdots \circ a_n}\right)^{-1} = a_n^{-1} \circ \cdots \circ a_2^{-1} \circ a_1^{-1}$

## Proof

 $\ds \paren {a \circ b} \circ \paren {b^{-1} \circ a^{-1} }$ $=$ $\ds \paren {\paren {a \circ b} \circ b^{-1} } \circ a^{-1}$ Definition of Associative Operation $\ds$ $=$ $\ds \paren {a \circ \paren {b \circ b^{-1} } } \circ a^{-1}$ Definition of Associative Operation $\ds$ $=$ $\ds \paren {a \circ e} \circ a^{-1}$ Behaviour of Inverse $\ds$ $=$ $\ds a \circ a^{-1}$ Behaviour of Identity $\ds$ $=$ $\ds e$ Behaviour of Inverse

Similarly for $\paren {b^{-1} \circ a^{-1} } \circ \paren {a \circ b}$.

$\blacksquare$