Inverse of Group Product

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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 $G \, 1$: Associativity
\(\displaystyle \) \(=\) \(\displaystyle \paren {a \circ \paren {b \circ b^{-1} } } \circ a^{-1}\) Group Axiom $G \, 1$: Associativity
\(\displaystyle \) \(=\) \(\displaystyle \paren {a \circ e} \circ a^{-1}\) Group Axiom $G \, 3$: Inverses
\(\displaystyle \) \(=\) \(\displaystyle a \circ a^{-1}\) Group Axiom $G \, 2$: Identity
\(\displaystyle \) \(=\) \(\displaystyle e\) Group Axiom $G \, 3$: Inverses

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

$\blacksquare$


Sources