After studying equations in groups, we now collect the main properties of inverses. These inverse laws are used constantly in group theory. They tell us that the identity is its own inverse, the inverse of an inverse returns the original element, and the inverse of a product reverses the order of the factors. The last rule is especially important in non-commutative groups, where reversing order cannot be ignored. In this lesson, students will prove the standard inverse properties and apply them to products of two or more group elements. :::theorem[Properties of Inverses] Let

(G,\circ)

be a group with identity element

e

. Then: (i)

e^{-1}=e

. (ii)

(a^{-1})^{-1}=a \quad \forall\, a\in G

. (iii)

(a\circ b)^{-1}=b^{-1}\circ a^{-1} \quad \forall\, a,b\in G

. ::: :::proof Given that

(G,\circ)

is a group with identity element

e

. To prove the stated properties of inverses. [1] To prove that

e^{-1}=e

. Since

e

is the identity element,

\begin{equation*} e\circ e=e. \end{equation*}

Thus

e

is an inverse element of

e

. Since the inverse of an element is unique,

\begin{equation*} e^{-1}=e. \end{equation*}

Therefore

e^{-1}=e

. [2] To prove that

(a^{-1})^{-1}=a \quad \forall\, a\in G

. Let

a\in G

. Then

\begin{equation*} a\circ a^{-1}=e \end{equation*}

and

\begin{equation*} a^{-1}\circ a=e. \end{equation*}

Therefore

a

is an inverse element of

a^{-1}

. Since the inverse of an element is unique,

\begin{equation*} (a^{-1})^{-1}=a. \end{equation*}

[3] To prove that

(a\circ b)^{-1}=b^{-1}\circ a^{-1} \quad \forall\, a,b\in G

. Let

a,b\in G

. Now

\begin{align*} (a\circ b)\circ(b^{-1}\circ a^{-1}) &=a\circ\big(b\circ(b^{-1}\circ a^{-1})\big) \quad [\because\, \circ \text{ is associative}]\\ &=a\circ\big((b\circ b^{-1})\circ a^{-1}\big) \quad [\because\, \circ \text{ is associative}]\\ &=a\circ(e\circ a^{-1}) \quad [\because\, b\circ b^{-1}=e]\\ &=a\circ a^{-1} \quad [\because\, e \text{ is the identity element}]\\ &=e. \end{align*}

Also

\begin{align*} (b^{-1}\circ a^{-1})\circ(a\circ b) &=b^{-1}\circ\big(a^{-1}\circ(a\circ b)\big) \quad [\because\, \circ \text{ is associative}]\\ &=b^{-1}\circ\big((a^{-1}\circ a)\circ b\big) \quad [\because\, \circ \text{ is associative}]\\ &=b^{-1}\circ(e\circ b) \quad [\because\, a^{-1}\circ a=e]\\ &=b^{-1}\circ b \quad [\because\, e \text{ is the identity element}]\\ &=e. \end{align*}

Therefore

b^{-1}\circ a^{-1}

is an inverse element of

a\circ b

. Since the inverse of an element is unique,

\begin{equation*} (a\circ b)^{-1}=b^{-1}\circ a^{-1}. \end{equation*}

Hence, all stated properties of inverses hold in

G

\square

::: The inverse of a product reverses the order of the factors. This reversal is easy to miss in commutative examples because the final numerical value may not change. Enter a list of symbols and compare the original product with the inverse product. The preview builds the formula by reading the symbols from right to left and attaching an inverse to each one. This is the same pattern used for products of two, three, or many group elements. :::scientific-preview[Reverse the Order Explorer]

Factors, separated by commas

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Abstract Algebra · Introduction to Groups

Properties of Inverses

Learn Properties of Inverses in Introduction to Groups.

Dr. Bivash Majumder Updated 26 Jun 2026 22 min

Understand the central mathematical ideas of Properties of Inverses.

Interpret the principal results and their mathematical conditions.

Follow and justify the main proof strategy step by step.

Apply the method to representative examples and problems.

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Elementary Properties of Groups

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Apply

Verify

\begin{equation*} e\circ e=e. \end{equation*}

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5 concepts

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4 guided steps

Applications

6 worked items

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1Introduce

2Visualize

3Derive

4Solve

5Practise

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Estimated time 22 min

Objectives5

Prerequisites0

1.Understand the central mathematical ideas of Properties of Inverses.
2.Interpret the principal results and their mathematical conditions.
3.Follow and justify the main proof strategy step by step.
4.Apply the method to representative examples and problems.
5.Practise the concept independently and verify the result.

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theorem

1. Properties of Inverses

Let $(G,\circ)$ be a group with identity element $e$ . Then: (i) $e^{-1}=e$ . (ii) $(a^{-1})^{-1}=a \quad \forall\, a\in G$ . (iii) $(a\circ b)^{-1}=b^{-1}\circ a^{-1} \quad \forall\, a,b\in G$ .

corollary

2. corollary

\begin{equation*} (a\circ b\circ c)^{-1}=c^{-1}\circ b^{-1}\circ a^{-1}. \end{equation*}

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Author: Dr. Bivash Majumder
Academic reviewer: Review not separately recorded
First published: 26 Jun 2026
Last reviewed: Not recorded
Last modified: 26 Jun 2026
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