# Category:Subgroups

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This category contains results about **Subgroups**.

Definitions specific to this category can be found in Definitions/Subgroups.

Let $\struct {G, \circ}$ be an algebraic structure.

$\struct {H, \circ}$ is a **subgroup** of $\struct {G, \circ}$ if and only if:

## Subcategories

This category has the following 29 subcategories, out of 29 total.

### C

- Cauchy's Group Theorem (3 P)
- Center of Group is Subgroup (3 P)
- Congruence Modulo Subgroup (8 P)

### E

### F

- Finite Subgroup Test (3 P)

### G

- Generated Normal Subgroups (empty)

### H

- Hall Subgroups (2 P)

### I

### L

### N

### O

- Order of Subgroup Product (6 P)

### S

- Subgroup Action (3 P)
- Subgroup Complements (3 P)
- Subset Product of Subgroups (9 P)

### T

- Tower Law for Subgroups (3 P)

### U

- Union of Subgroups (5 P)

## Pages in category "Subgroups"

The following 81 pages are in this category, out of 81 total.

### C

- Cauchy's Group Theorem
- Cayley's Representation Theorem
- Center of Group is Abelian Subgroup
- Center of Group is Subgroup
- Centralizer in Subgroup is Intersection
- Centralizer of Group Element is Subgroup
- Condition for Elements of Group to be in Subgroup
- Condition for Partition between Invertible and Non-Invertible Elements to induce Congruence Relation on Monoid
- Conjugate of Subgroup is Subgroup

### E

### G

- Gaussian Integers form Subgroup of Complex Numbers under Addition
- Group does not Necessarily have Subgroup of Order of Divisor of its Order
- Group Epimorphism Preserves Subgroups
- Group has Subgroups of All Prime Power Factors
- Group Homomorphism Preserves Subgroups
- Group is Subgroup of Itself
- Group with Normal Series with Solvable Factor Groups is Solvable

### I

- Identity of Subgroup
- Image of Group Homomorphism is Subgroup
- Index in Subgroup
- Index of Intersection of Subgroups
- Index of Subgroup equals Index of Conjugate
- Indicator is Well-Defined
- Infimum and Supremum of Subgroups
- Infimum of Subgroups in Lattice
- Intersection of Subgroups is Subgroup
- Intersection of Subgroups is Subgroup/General Result
- Intersection of Subgroups of Prime Order
- Intersection with Subgroup Product of Superset
- Inverse of Subgroup
- Inverses in Subgroup
- Invertible Elements of Monoid form Subgroup

### O

### P

- Permutation Group is Subgroup of Symmetric Group
- Power of Element in Subgroup
- Powers of Element form Subgroup
- Preimage of Image of Subgroup under Group Epimorphism
- Preimage of Subgroup under Epimorphism is Subgroup
- Prime Group has no Proper Subgroups
- Product of Subgroup with Inverse
- Product of Subgroup with Itself
- Product of Subgroups of Prime Power Order
- Pullback of Quotient Group Isomorphism is Subgroup

### S

- Set of Homomorphisms to Abelian Group is Subgroup of All Mappings
- Set of Subgroups forms Complete Lattice
- Strictly Positive Rational Numbers under Multiplication form Subgroup of Non-Zero Rational Numbers
- Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers
- Subgroup Generated by Subgroup and Element
- Subgroup is Normal Subgroup of Normalizer
- Subgroup is Subgroup of Normalizer
- Subgroup of Abelian Group is Abelian
- Subgroup of Additive Group Modulo m is Ideal of Ring
- Subgroup of Cyclic Group is Cyclic
- Subgroup of Index 2 contains all Squares of Group Elements
- Subgroup of Index 3 does not necessarily contain all Cubes of Group Elements
- Subgroup of Infinite Cyclic Group is Infinite Cyclic Group
- Subgroup of Integers is Ideal
- Subgroup of Order 1 is Trivial
- Subgroup of Solvable Group is Solvable
- Subgroup Subset of Subgroup Product
- Subgroups of Additive Group of Integers
- Subset Product of Subgroups
- Supremum of Subgroups in Lattice
- Sylow Theorems