Category:Subgroup property implications
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 210 total. The count includes redirect pages that have been included in the category. Redirect pages are shown in italics.
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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
- Powering-invariant subgroup
- 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