Abstract AlgebraIntroduction to GroupsElementary Properties of Groups
Inverse Element in Groups
After proving that the identity element of a group is unique, we next prove that the inverse of each element is also unique. This is one of the most frequently used facts in group theory. The group definition says that for each a∈G, there exists an inverse element a−1∈G, but the definition does not initially say that the inverse is the only one. The uniqueness theorem allows us to write a−1 without ambiguity. In this lesson, students will learn the formal proof of uniqueness of inverses, how inverse notation changes in additive and multiplicative groups, and why both left and right inverse equations are essential.
:::definition[Inverse Element]
Let (G,∘) be a group with identity element e, and let a∈G. An element b∈G is called an inverseelement of a if
a∘b=b∘a=e.
When the inverse element is unique, it is denoted by a−1.
:::
An inverse element depends on the element being inverted. In (Z,+), the inverse of 7 is −7, and the inverse of −3 is 3. In (R−{0},⋅), the inverse of 5 is 51, and the inverse of −32 is −23. The identity element is common to all elements, but inverses vary from element to element. This distinction is a common source of confusion in early group theory.
:::theorem
Let (G,∘) be a group with identity element e and let a∈G. Then the inverse of a in G is unique.
:::
:::proof
Given that (G,∘) is a group with identity element e and a∈G.
To prove that the inverse of a in G is unique.
Let b,c∈G be two inverses of a.
Then
a∘b=b∘a=e
and
a∘c=c∘a=e.
Now
b=b∘e[∵e is the identity element]=b∘(a∘c)[∵a∘c=e]=(b∘a)∘c[∵∘ is associative]=e∘c[∵b∘a=e]=c[∵e is the identity element].
Therefore b=c.
Hence, the inverse of a in G is unique.
□
:::
An inverse candidate must work from both sides. In additive modular arithmetic, every element has a unique inverse, but in multiplication modulo n some elements may fail to have an inverse on the whole residue set. Choose an element and compare every possible candidate. A candidate is accepted only when both products equal the identity. This makes the two-sided inverse condition visible and helps separate group examples from non-group examples.
:::scientific-preview[Inverse Pair Explorer]
Inverse Element in Groups | Introduction to Groups | BMLabs | Introduction to Groups | BMLabs Mathematics | BMLabs Mathematics
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Abstract Algebra · Introduction to Groups
Inverse Element in Groups
Learn Inverse Element in Groups in Introduction to Groups.
After proving that the identity element of a group is unique, we next prove that the inverse of each element is also unique. This is one of the most frequently used facts in group theory. The group definition says that for each a∈G, there exists an inverse element a−1∈G, but the definition does not initially say that the inverse is the only one. The uniqueness theorem allows us to write a−1 without ambiguity. In this lesson, students will learn the formal proof of uniqueness of inverses, how inverse notation changes in additive and multiplicative groups, and why both left and right inverse equations are essential.
Core definition02
Inverse Element
Let (G,∘) be a group with identity element e, and let a∈G. An element b∈G is called an inverseelement of a if
a∘b=b∘a=e.
When the inverse element is unique, it is denoted by a−1.
An inverse element depends on the element being inverted. In (Z,+), the inverse of 7 is −7, and the inverse of −3 is 3. In (R−{0},⋅), the inverse of 5 is 51, and the inverse of −32 is −23. The identity element is common to all elements, but inverses vary from element to element. This distinction is a common source of confusion in early group theory.
Key result04
Let (G,∘) be a group with identity element e and let a∈G. Then the inverse of a in G is unique.
Reasoning pathway05
Given that (G,∘) is a group with identity element e and a∈G.
To prove that the inverse of a in G is unique.
Let b,c∈G be two inverses of a.
Then
a∘b=b∘a=e
and
a∘c=c∘a=e.
Now
b=b∘e[∵e is the identity element]=b∘(a∘c)[∵a∘c=e]=(b∘a)∘c[∵∘ is associative]=e∘c[∵b∘a=e]=c[∵e is the identity element].
Therefore b=c.
Hence, the inverse of a in G is unique.
□
An inverse candidate must work from both sides. In additive modular arithmetic, every element has a unique inverse, but in multiplication modulo n some elements may fail to have an inverse on the whole residue set. Choose an element and compare every possible candidate. A candidate is accepted only when both products equal the identity. This makes the two-sided inverse condition visible and helps separate group examples from non-group examples.
Visual laboratory
Inverse Pair Explorer
INVERSE PAIR EXPLORER
Dynamic Sandbox
Initializing Workspace
The proof uses associativity in the middle step. This is why uniqueness of inverse is a group property and not merely a property of a set with an identity. The expression b∘(a∘c) is changed to (b∘a)∘c, and then the two inverse assumptions are used. Students often try to cancel a before cancellation laws have been proved. At this point, cancellation should not be used as a shortcut because cancellation itself is proved later from inverses.
Guided example09
In (Z,+), the inverse of a∈Z is −a.
Indeed,
a+(−a)=(−a)+a=0.
By the uniqueness of inverse, no other integer can be the additive inverse of a. For example, the inverse of 9 is −9 only.
Guided example10
In (Q−{0},⋅), the inverse of a∈Q−{0} is a1.
Indeed,
a⋅a1=a1⋅a=1.
By the uniqueness of inverse, no other nonzero rational number can be the multiplicative inverse of a.
Worked problem11
Let (G,∘) be a group with identity element e. Suppose a,b∈G and
a∘b=e,b∘a=e.
Prove that b=a−1.
Complete solution12
Let (G,∘) be a group with identity element e, and let a,b∈G.
Given that
a∘b=e,b∘a=e.
To prove that b=a−1.
Since
a∘b=b∘a=e,
b is an inverse element of a.
Since the inverse of a is unique, the inverse element of a is denoted by a−1.
Therefore
b=a−1.
Hence, b=a−1.
□
Worked problem13
Let (G,∘) be a group with identity element e. If a∈G and a∘a=e, prove that a−1=a.
Complete solution14
Let (G,∘) be a group with identity element e and let a∈G.
Given that
a∘a=e.
To prove that a−1=a.
Since a∘a=e, and the two factors are both a, we also have
a∘a=e.
Thus a is an inverse element of a.
Since the inverse of a is unique, we get
a−1=a.
Hence, a−1=a.
□
Mathematical insight15
The condition a∘a=e says that a is its own inverse. Such elements occur often. In the multiplicative group {1,−1}, both elements are their own inverses. In the additive group (Z,+), only 0 is its own inverse, because a+a=0 gives a=0 in integers. The statement must always be interpreted relative to the operation of the group.
Use the calculator below to test inverse candidates in modular addition or modular multiplication. Choose a modulus n, an element a, and a candidate b. The calculator checks whether a∘b=b∘a=e for the selected operation. This helps distinguish one-sided computation from the two-sided inverse condition in a group.
Interactive calculator
Inverse Test
INVERSE TEST
Initializing Workspace
Independent practice18
[1] Find the inverse of 12 in (Z,+).
[2] Find the inverse of 75 in (Q−{0},⋅).
[3] Let (G,∘) be a group with identity element e. If a∘b=b∘a=e, prove that a=b−1.
[4] Let (G,∘) be a group. If a−1=b−1, prove that a=b.
[5] Give an example of an element that is its own inverse.
Answer19
[1] The inverse is −12.
[2] The inverse is 57.
[3] Since a∘b=b∘a=e, a is an inverse of b. By uniqueness of inverse, a=b−1.
[4] Taking inverses on both sides gives (a−1)−1=(b−1)−1, hence a=b.
[5] In (Z,+), the element 0 is its own inverse. In (R−{0},⋅), the element −1 is its own inverse.
Questions to consolidate
Frequently asked questions
4
1Does every element of a group have an inverse?
Yes. The inverse axiom requires every element of a group to have an inverse in the group.
2Can an element have two different inverses?
No. The inverse of each element in a group is unique.
3Is the inverse always written as a−1?
In multiplicative notation yes. In additive notation, the inverse of a is usually written as −a.
4Why is associativity used in the proof of uniqueness?
Associativity allows the expression b∘(a∘c) to be rewritten as (b∘a)∘c.
Continue learning
Cancellation Laws
Continue to cancellation laws, where inverse elements are used to cancel common factors in group equations.
