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

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

\(\ds \paren {a \circ b} \circ \paren {a \circ b}^{-1}\) \(=\) \(\ds e\) Definition of Inverse Element
\(\ds \leadsto \ \ \) \(\ds a \circ \paren {b \circ \paren {a \circ b}^{-1} }\) \(=\) \(\ds e\) Group Axiom $\text G 1$: Associativity
\(\ds \leadsto \ \ \) \(\ds b \circ \paren {a \circ b}^{-1}\) \(=\) \(\ds a^{-1}\) Group Product Identity therefore Inverses
\(\ds \leadsto \ \ \) \(\ds b^{-1} \circ \paren {b \circ \paren {a \circ b}^{-1} }\) \(=\) \(\ds b^{-1} \circ a^{-1}\) Group Axiom $\text G 3$: Existence of Inverse Element
\(\ds \leadsto \ \ \) \(\ds \paren{b^{-1} \circ b} \circ \paren {a \circ b}^{-1}\) \(=\) \(\ds b^{-1} \circ a^{-1}\) Group Axiom $\text G 1$: Associativity
\(\ds \leadsto \ \ \) \(\ds e \circ \paren {a \circ b}^{-1}\) \(=\) \(\ds b^{-1} \circ a^{-1}\) Definition of Inverse Element
\(\ds \leadsto \ \ \) \(\ds \paren {a \circ b}^{-1}\) \(=\) \(\ds b^{-1} \circ a^{-1}\) Definition of Identity Element

$\blacksquare$


Sources