# Category:Group Theory

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This category contains results about **Group Theory**.

Definitions specific to this category can be found in Definitions/Group Theory.

**Group Theory** is a branch of abstract algebra which studies groups and other related algebraic structures.

## Subcategories

This category has the following 99 subcategories, out of 99 total.

### A

- Abnormal Subgroups (1 P)
- Ambivalent Groups (2 P)

### B

### C

- Cancellation Laws (14 P)
- Characteristic Subgroups (3 P)
- Congruence Modulo Subgroup (8 P)

### D

- Decomposable Groups (2 P)

### E

- Examples of Words (3 P)

### F

- Fedorov Groups (2 P)
- Free Groups (empty)
- Friezes (1 P)

### G

- Group Presentations (7 P)
- Group Rings (1 P)
- Group Words (3 P)

### H

- Hamiltonian Groups (1 P)

### I

- Identity of Group is Unique (4 P)
- Inverse in Group is Unique (4 P)
- Inverse of Commuting Pair (3 P)
- Inverse of Group Inverse (4 P)
- Inverse of Group Product (7 P)
- Isomorphism Preserves Groups (3 P)

### L

- Lie Groups (empty)

### M

### N

### O

### P

- Paranormal Subgroups (empty)
- Polynormal Subgroups (empty)
- Product Inverse Operation (11 P)
- Pronormal Subgroups (empty)

### Q

### R

### S

- Self-Inverse Elements (3 P)
- Self-Normalizing Subgroups (1 P)
- Set of Words Generates Group (2 P)
- Sporadic Groups (1 P)
- Sporadic Simple Groups (empty)

### T

- Tessellations (5 P)

### W

- Wallpaper Groups (1 P)
- Wallpaper Patterns (1 P)
- Weakly Abnormal Subgroups (empty)
- Weakly Pronormal Subgroups (empty)

## Pages in category "Group Theory"

The following 98 pages are in this category, out of 98 total.

### C

- Cancellation Laws
- Cauchy's Lemma (Group Theory)
- Cayley's Representation Theorem
- Cayley's Representation Theorem/General Case
- Center of Group of Prime Power Order is Non-Trivial
- Centralizer of Subset is Intersection of Centralizers of Elements
- Character of Representations over C are Algebraic Integers
- Commutation Property in Group
- Commutation with Group Elements implies Commuation with Product with Inverse
- Commutativity of Powers in Group
- Complement of Relation Compatible with Group is Compatible
- Condition for Group given Semigroup with Idempotent Element
- Conditions under which Commutative Semigroup is Group

### E

- Element Commutes with Square in Group
- Element to Power of Group Order is Identity
- Element to Power of Remainder
- Equivalence of Axiom Schemata for Groups
- Equivalence of Axiom Schemata for Groups/Warning
- Equivalence of Definitions of Direct Limit of Sequence of Groups
- Equivalence of Definitions of Normal Subset
- Existence and Uniqueness of Direct Limit of Sequence of Groups
- Existence of Unique Subgroup Generated by Subset
- External Direct Product of Groups is Group/Finite Product

### G

- General Morphism Property for Groups
- Group Direct Product is Product in Category of Groups
- Group Element Commutes with Inverse
- Group has Latin Square Property
- Group has Latin Square Property/Corollary
- Group Induced by B-Algebra Induced by Group
- Group Induces B-Algebra
- Group is B-Algebra Iff All Elements Self-Inverse
- Group is Cancellable Monoid
- Group is Inverse Semigroup with Identity
- Group is not Empty
- Group is Quasigroup
- Group Operation is Cancellable
- Group Product Identity therefore Inverses
- Group with Zero Element is Trivial

### I

- Identity is only Idempotent Element in Group
- Identity of Group is Unique
- Identity of Subsemigroup of Group
- Inverse in Group is Unique
- Inverse of Commuting Pair
- Inverse of Element in Semidirect Product
- Inverse of Group Inverse
- Inverse of Group Product
- Inverse of Group Product/General Result
- Inverse of Inverse of Subset of Group
- Inverse of Product of Subsets of Group
- Invertible Elements of Monoid form Subgroup of Cancellable Elements
- Isomorphism Preserves Groups

### P

- Positive-Term Generalized Sum Converges iff Supremum
- Power of Product of Commutative Elements in Group
- Power of Product with Inverse
- Power Structure of Group is Monoid
- Power Structure of Group is Semigroup
- Powers of Commutative Elements in Groups
- Powers of Group Element Commute
- Powers of Group Elements
- Powers of Group Elements/Negative Index
- Powers of Group Elements/Negative Index/Additive Notation
- Powers of Group Elements/Product of Indices
- Powers of Group Elements/Product of Indices/Additive Notation
- Powers of Group Elements/Sum of Indices
- Powers of Group Elements/Sum of Indices/Additive Notation
- Product of Powers of Group Elements
- Product with Inverse equals Identity iff Equality
- Properties of Relation Compatible with Group Operation
- Pullback is Subgroup

### R

### S

- Schur-Zassenhaus Theorem
- Self-Inverse Elements Commute iff Product is Self-Inverse
- Semigroup is Group Iff Latin Square Property Holds
- Set Equivalence of Regular Representations
- Set of Words Generates Group
- Socks-Shoes Property
- Structure Induced by Group Operation is Group
- Structure is Group iff Semigroup and Quasigroup
- Subgroup of Real Numbers is Discrete or Dense
- Subset Product is Subset of Generator
- Sum of Powers of Group Elements