# Category:Inverse Elements

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This category contains results about **Inverse Elements** in the context of **Abstract Algebra**.

Definitions specific to this category can be found in **Definitions/Inverse Elements**.

The element $y$ is an **inverse of $x$** if and only if:

- $y \circ x = e_S = x \circ y$

that is, if and only if $y$ is both:

- a left inverse of $x$

and:

- a right inverse of $x$.

## Subcategories

This category has the following 12 subcategories, out of 12 total.

### E

### G

### I

- Inverse in Group is Unique (4 P)
- Inverse of Group Inverse (4 P)
- Inverse of Group Product (7 P)

### S

- Self-Inverse Elements (3 P)

## Pages in category "Inverse Elements"

The following 38 pages are in this category, out of 38 total.

### C

- Commutation of Inverses in Monoid
- Commutation with Inverse in Monoid
- Commutator of Group Element with Identity is Identity
- Condition for Invertibility in Power Structure on Associative or Cancellable Operation
- Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid

### E

### I

- Inverse Element is Power of Order Less 1
- Inverse in Group is Unique
- Inverse in Monoid is Unique
- Inverse not always Unique for Non-Associative Operation
- Inverse of Field Product
- Inverse of Group Commutator
- Inverse of Group Inverse
- Inverse of Group Product
- Inverse of Identity Element is Itself
- Inverse of Inverse
- Inverse of Inverse in General Algebraic Structure
- Inverse of Inverse in Monoid
- Inverse of Inverse/General Algebraic Structure
- Inverse of Inverse/Monoid
- Inverse of Product
- Inverse of Product in Associative Structure
- Inverse of Product in Monoid
- Inverse of Product/Monoid
- Inverse of Product/Monoid/General Result
- Invertible Element containing Identity in Power Structure
- Invertible Element of Associative Structure is Cancellable
- Invertible Elements of Monoid form Subgroup
- Isomorphism Preserves Inverses