Inverse of an Inverse

Theorem

General Algebraic Structures

Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $x \in S$ be invertible, and let $y$ be an inverse of $x$.

Then $x$ is also an inverse of $y$.

Monoids

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

Let $x \in S$ be invertible, and let its inverse be $x^{-1}$.

Then $x^{-1}$ is also invertible, and:

$\left({x^{-1}}\right)^{-1} = x$

Proof

Algebraic Structure

Let $x \in S$ be invertible, where $y$ is an inverse of $x$.

Then:

$x \circ y = e = y \circ x$

by definition.

Proof for Monoid

If $\left({S, \circ}\right)$ is a monoid then by definition $\circ$ is associative.

So any inverse of $x$ is unique, and can be denoted $x^{-1}$.

From the result for algebraic structures, $x^{-1}$ is also invertible and its inverse is $x$.

Thus we see that $\left({x^{-1}}\right)^{-1} = x$.

$\blacksquare$

Proof for Group

For use when $G$ is a group.

Let $g \in G$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle g\)	\(\in\)	\(\displaystyle G\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle e\)	\(=\)	\(\displaystyle g^{-1}g\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of inverse
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle (g^{-1})^{-1}e\)	\(=\)	\(\displaystyle ((g^{-1})^{-1}(g^{-1}g)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle (g^{-1})^{-1}e\)	\(=\)	\(\displaystyle ((g^{-1})^{-1}g^{-1})g\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle (g^{-1})^{-1}e\)	\(=\)	\(\displaystyle eg\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of inverse
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle (g^{-1})^{-1}\)	\(=\)	\(\displaystyle g\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of identity

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.6$: Theorem $\text{(iii)}$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 4$: Theorem $4.3$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.4$: Theorem $2 \ \text{(i)}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 28 \ (2)$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 35.5$
• John F. Humphreys: A Course in Group Theory (1996): $\S 3$: Corollary $3.5$