# Category:Subgroup property implications

From Groupprops

This category lists important implications between properties over the context space: subgroup. That is, it says that every subgroup satisfying the first subgroup property also satisfies the second

View other property implication categories

A related category is Category:Subgroup property non-implications

Also refer Category:Composition computations

## Subcategories

This category has only the following subcategory.

## Pages in category "Subgroup property implications"

The following 200 pages are in this category, out of 209 total. The count *includes* redirect pages that have been included in the category. Redirect pages are shown in italics.

### 2

### A

- Abelian and pronormal implies SCDIN
- Abelian direct factor implies potentially verbal in finite
- Abelian implies every subgroup is potentially characteristic
- Abnormal implies WNSCC
- Amalgam-characteristic implies image-potentially characteristic
- Amalgam-characteristic implies potentially characteristic
- Artinian implies periodic

### B

### C

- C-closed implies completely divisibility-closed in nilpotent group
- C-closed implies local powering-invariant
- C-closed implies powering-invariant
- CDIN of conjugacy-closed implies CDIN
- Center of pronormal implies SCDIN
- Central factor implies normal
- Central factor implies transitively normal
- Central implies abelian normal
- Central implies amalgam-characteristic
- Central implies finite-pi-potentially verbal in finite
- Central implies image-potentially characteristic
- Central implies normal
- Central implies normal satisfying the subgroup-to-quotient powering-invariance implication
- Central implies potentially characteristic
- Central implies potentially fully invariant in finite
- Central implies potentially verbal in finite
- Central subgroup implies join-transitively central factor
- CEP implies every relatively normal subgroup is weakly closed
- Characteristic and self-centralizing implies coprime automorphism-faithful
- Characteristic central factor of WNSCDIN implies WNSCDIN
- Characteristic equals fully invariant in odd-order abelian group
- Characteristic equals strictly characteristic in Hopfian
- Characteristic equals verbal in free abelian group
- Characteristic implies automorph-conjugate
- Characteristic implies normal
- Characteristic of CDIN implies CDIN
- Characteristic subgroup of abelian group implies intermediately powering-invariant
- Characteristic subgroup of abelian group implies powering-invariant
- Characteristic subgroup of abelian group is quotient-powering-invariant
- Characteristic upper-hook AEP implies characteristic
- Cocentral implies central factor
- Cocentral implies centralizer-dense
- Cocentral implies right-quotient-transitively central factor
- Cofactorial automorphism-invariant implies left-transitively 2-subnormal
- Commutator of a group and a subgroup implies normal
- Commutator of a normal subgroup and a subset implies 2-subnormal
- Commutator of a transitively normal subgroup and a subset implies normal
- Commutator-verbal implies divisibility-closed in nilpotent group
- Comparable with all normal subgroups implies characteristic in finite nilpotent group
- Comparable with all normal subgroups implies normal in nilpotent group
- Complemented normal implies endomorphism kernel
- Conjugacy-closed implies focal subgroup equals derived subgroup
- Conjugate-join-closed subnormal implies join-transitively subnormal
- Conjugate-permutable implies subnormal in finite
- Cyclic characteristic implies hereditarily characteristic
- Cyclic normal implies finite-pi-potentially verbal in finite
- Cyclic normal implies hereditarily normal
- Cyclic normal implies potentially verbal in finite
- Cyclic-quotient characteristic implies upward-closed characteristic

### D

- Direct factor implies central factor
- Direct factor implies join-transitively central factor
- Direct factor implies normal
- Direct factor implies right-quotient-transitively central factor
- Direct factor implies transitively normal
- Divisibility-closed implies powering-invariant
- Double coset index two implies maximal

### E

### F

- Finite implies local powering-invariant
- Finite implies powering-invariant
- Finite index implies completely divisibility-closed
- Finite index implies powering-invariant
- Finite normal implies amalgam-characteristic
- Finite normal implies image-potentially characteristic
- Finite normal implies quotient-powering-invariant
- Finite solvable-extensible implies class-preserving
- Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups
- Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
- Focal subgroup theorem
- Free factor implies self-normalizing or trivial
- Fully invariant direct factor implies left-transitively homomorph-containing
- Fully invariant implies characteristic
- Fully invariant implies finite direct power-closed characteristic
- Fully invariant implies verbal in reduced free group
- Fully invariant of strictly characteristic implies strictly characteristic

### H

- Hall implies join of Sylow subgroups
- Hall implies order-conjugate in solvable
- Hall implies order-dominating in finite solvable
- Hall implies paracharacteristic
- Hall retract implies order-conjugate
- Hall-extensible implies class-preserving
- Homocyclic normal implies finite-pi-potentially fully invariant in finite
- Homocyclic normal implies potentially fully invariant in finite

### I

- Identity functor controls strong fusion for abelian Sylow subgroup
- Index four implies 2-subnormal or double coset index two
- Index three implies normal or double coset index two
- Intermediately automorph-conjugate implies intermediately characteristic in nilpotent
- Intermediately automorph-conjugate of normal implies weakly pronormal
- Intermediately isomorph-conjugate of normal implies pronormal
- Intermediately normal-to-characteristic implies intermediately characteristic in nilpotent
- Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
- Intersection of DPICF and direct factor is central factor
- Intersection of kernels of bihomomorphisms implies completely divisibility-closed
- Isomorph-containing implies characteristic
- Isomorph-normal coprime automorphism-invariant of Sylow implies weakly closed
- Iterated agemo implies verbal

### J

### K

### L

### M

- Marginal implies direct power-closed characteristic
- Marginal implies unconditionally closed
- Maximal among abelian normal implies self-centralizing in nilpotent
- Maximal among abelian normal implies self-centralizing in supersolvable
- Maximal implies modular
- Maximal implies pronormal
- Minimal characteristic implies central in nilpotent
- Minimal normal implies central in nilpotent group
- Minimal normal implies powering-invariant in solvable group
- Monolith is characteristic
- Monolith is strictly characteristic

### N

- NE implies weakly normal
- Nilpotent Hall implies isomorph-conjugate
- Nilpotent implies every maximal subgroup is normal
- Nilpotent implies every normal subgroup is potentially characteristic
- Nilpotent-quotient implies subgroup-to-quotient powering-invariance implication
- Normal and centralizer-free implies automorphism-faithful
- Normal and self-centralizing implies coprime automorphism-faithful
- Normal and self-centralizing implies normality-large
- Normal Hall implies permutably complemented
- Normal implies join-transitively subnormal
- Normal implies permutable
- Normal of finite index implies completely divisibility-closed
- Normal of finite index implies quotient-powering-invariant
- Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication
- Normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication
- Normal-homomorph-containing implies strictly characteristic
- Normal-potentially characteristic implies normal-extensible automorphism-invariant
- Normality-preserving endomorphism-invariant implies characteristic
- Normality-preserving endomorphism-invariant implies finite direct power-closed characteristic
- Normalizer of intermediately subnormal-to-normal implies self-normalizing
- Normalizer of pronormal implies abnormal

### P

- Paranormal implies polynormal
- Paranormal implies weakly normal
- Perfect subnormal implies join-transitively subnormal
- Perfect subnormal implies subnormal-permutable
- Periodic normal implies amalgam-characteristic
- Periodic normal implies image-potentially characteristic
- Periodic normal implies potentially characteristic
- Permutable and subnormal implies join-transitively subnormal
- Permutable implies ascendant
- Permutable implies locally subnormal
- Permutable implies modular
- Permutable implies subnormal in finite
- Polynormal implies intermediately subnormal-to-normal
- Potentially characteristic implies normal
- Potentially verbal implies normal
- Prehomomorph-contained implies strictly characteristic
- Prime power order implies center is normality-large
- Pronormal and subnormal implies normal
- Pronormal implies intermediately subnormal-to-normal
- Pronormal implies MWNSCDIN
- Pronormal implies self-conjugate-permutable
- Pronormal implies weakly normal
- Pronormal implies WNSCDIN

### R

### S

- SCDIN of subset-conjugacy-closed implies SCDIN
- Self-centralizing and minimal normal implies characteristic
- Self-centralizing and minimal normal implies strictly characteristic
- Strongly paranormal implies paranormal
- Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal
- Subgroup of finite group has a left transversal that is also a right transversal
- Subgroup of finite index has a left transversal that is also a right transversal
- Subgroup of index equal to least prime divisor of group order is normal
- Subgroup of index two is normal
- Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal
- Subhomomorph-containing implies right-transitively homomorph-containing
- Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup
- Sylow and TI implies CDIN
- Sylow and TI implies SCDIN
- Sylow implies automorph-conjugate
- Sylow implies intermediately isomorph-conjugate
- Sylow implies isomorph-conjugate
- Sylow implies MWNSCDIN
- Sylow implies order-conjugate
- Sylow implies order-dominated
- Sylow implies order-dominating