# Inverse of Group Product

## Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $a, b \in G$, with inverses $a^{-1}, b^{-1}$.

Then:

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

### General Result

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $a_1, a_2, \ldots, a_n \in G$, with inverses $a_1^{-1}, a_2^{-1}, \ldots, a_n^{-1}$.

Then:

$\paren {a_1 \circ a_2 \circ \cdots \circ a_n}^{-1} = a_n^{-1} \circ \cdots \circ a_2^{-1} \circ a_1^{-1}$

## Proof 1

 $\displaystyle \paren {a \circ b} \circ \paren {b^{-1} \circ a^{-1} }$ $=$ $\displaystyle \paren {\paren {a \circ b} \circ b^{-1} } \circ a^{-1}$ Group Axiom $\text G 1$: Associativity $\displaystyle$ $=$ $\displaystyle \paren {a \circ \paren {b \circ b^{-1} } } \circ a^{-1}$ Group Axiom $\text G 1$: Associativity $\displaystyle$ $=$ $\displaystyle \paren {a \circ e} \circ a^{-1}$ Group Axiom $\text G 3$: Existence of Inverse Element $\displaystyle$ $=$ $\displaystyle a \circ a^{-1}$ Group Axiom $\text G 2$: Existence of Identity Element $\displaystyle$ $=$ $\displaystyle e$ Group Axiom $\text G 3$: Existence of Inverse Element

The result follows from Group Product Identity therefore Inverses:

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

$\blacksquare$

## Proof 2

We have that a group is a monoid, all of whose elements are invertible.

The result follows from Inverse of Product in Monoid.

$\blacksquare$

## Proof 3

 $\displaystyle \paren {a \circ b} \circ \paren {a \circ b}^{-1}$ $=$ $\displaystyle e$ Definition of Inverse Element $\displaystyle \leadsto \ \$ $\displaystyle a \circ \paren {b \circ \paren {a \circ b}^{-1} }$ $=$ $\displaystyle e$ Group Axiom $G1$: Associativity $\displaystyle \leadsto \ \$ $\displaystyle b \circ \paren {a \circ b}^{-1}$ $=$ $\displaystyle a^{-1}$ Group Product Identity therefore Inverses $\displaystyle \leadsto \ \$ $\displaystyle b^{-1} \circ b \circ \paren {a \circ b}^{-1}$ $=$ $\displaystyle b^{-1} \circ a^{-1}$ $\displaystyle \leadsto \ \$ $\displaystyle e \circ \paren {a \circ b}^{-1}$ $=$ $\displaystyle b^{-1} \circ a^{-1}$ Definition of Inverse Element $\displaystyle \leadsto \ \$ $\displaystyle \paren {a \circ b}^{-1}$ $=$ $\displaystyle b^{-1} \circ a^{-1}$ Definition of Identity Element

$\blacksquare$