# Category:Subgroup property non-implications

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This category lists important non-implications between properties over the context space: subgroup. That is, it says that every subgroup satisfying the first subgroup property need not satisfy the second.

View other property non-implication categories

## Pages in category "Subgroup property non-implications"

The following 150 pages are in this category, out of 150 total. The count *includes* redirect pages that have been included in the category. Redirect pages are shown in italics.

### A

### C

- Central factor not implies direct factor
- Characteristic direct factor not implies fully invariant
- Characteristic not implies amalgam-characteristic
- Characteristic not implies characteristic-isomorph-free in finite
- Characteristic not implies direct factor
- Characteristic not implies elementarily characteristic
- Characteristic not implies fully invariant
- Characteristic not implies fully invariant in finite abelian group
- Characteristic not implies fully invariant in finitely generated abelian group
- Characteristic not implies fully invariant in odd-order class two p-group
- Characteristic not implies injective endomorphism-invariant
- Characteristic not implies injective endomorphism-invariant in finitely generated abelian group
- Characteristic not implies isomorph-free in finite group
- Characteristic not implies isomorph-normal in finite group
- Characteristic not implies normal-isomorph-free
- Characteristic not implies potentially fully invariant
- Characteristic not implies powering-invariant in nilpotent group
- Characteristic not implies powering-invariant in solvable group
- Characteristic not implies quasiautomorphism-invariant
- Characteristic not implies strictly characteristic
- Characteristic not implies sub-(isomorph-normal characteristic) in finite
- Characteristic not implies sub-isomorph-free in finite group
- Characteristic subgroup of abelian group not implies divisibility-closed
- Characteristic subgroup of abelian group not implies local powering-invariant
- Characteristic-isomorph-free not implies normal-isomorph-free in finite
- Cocentral not implies amalgam-characteristic
- Complemented central factor not implies direct factor
- Complemented characteristic not implies left-transitively complemented normal
- Complemented normal not implies direct factor
- Complemented normal not implies local powering-invariant
- Conjugacy-closed and Hall not implies retract
- Conjugacy-closed normal not implies central factor
- Conjugacy-closed not implies weak subset-conjugacy-closed
- Conjugate-comparable not implies normal
- Contranormal not implies self-normalizing

### D

### E

### F

- Finite direct power-closed characteristic not implies fully invariant
- Finite index not implies local powering-invariant
- Finite not implies divisibility-closed in abelian group
- Finite solvable not implies p-normal
- Fully invariant not implies abelian-potentially verbal in abelian group
- Fully invariant not implies verbal in finite abelian group
- Fully invariant subgroup of abelian group not implies divisibility-closed
- Fusion system-relatively weakly closed not implies isomorph-normal

### H

- Hall not implies automorph-conjugate
- Hall not implies order-conjugate
- Hall not implies order-isomorphic
- Hall not implies procharacteristic
- Hall not implies pronormal
- Hall not implies WNSCDIN
- Hereditarily characteristic not implies cyclic in finite
- Homomorph-containing not implies no nontrivial homomorphism to quotient group

### I

- Image-closed characteristic not implies fully invariant
- Image-closed fully invariant not implies verbal
- Index two not implies characteristic
- Intermediately characteristic not implies isomorph-containing in abelian group
- Intermediately characteristic not implies isomorph-containing in group of prime power order
- Iterated agemo subgroup not implies agemo subgroup

### L

- Lattice-complemented not implies permutably complemented
- Left-transitively homomorph-containing not implies subhomomorph-containing
- Left-transitively WNSCDIN not implies characteristic
- Left-transitively WNSCDIN not implies normal
- Local powering-invariant not implies local divisibility-closed
- Locally inner automorphism-balanced not implies central factor

### M

### N

- Nilpotent not implies ACIC
- No common composition factor with quotient group not implies complemented
- No nontrivial homomorphism from quotient group not implies characteristic
- No nontrivial homomorphism to quotient group not implies complemented normal
- Normal not implies amalgam-characteristic
- Normal not implies central factor
- Normal not implies characteristic
- Normal not implies direct factor
- Normal not implies finite-pi-potentially characteristic in finite
- Normal not implies image-potentially fully invariant
- Normal not implies left-transitively fixed-depth subnormal
- Normal not implies normal-extensible automorphism-invariant in finite
- Normal not implies normal-potentially characteristic
- Normal not implies normal-potentially relatively characteristic
- Normal not implies potentially fully invariant
- Normal not implies potentially verbal
- Normal not implies right-transitively fixed-depth subnormal
- Normal not implies strongly potentially characteristic
- Normal-extensible not implies inner
- Normal-extensible not implies normal
- Normal-isomorph-free not implies isomorph-free in finite
- Normal-potentially characteristic not implies characteristic

### O

### P

- P-automorphism-invariant not implies characteristic
- P-normal not implies p-solvable
- Panferov Lie group for 5
- Paranormal not implies pronormal
- Permutable not implies normal
- Potentially characteristic not implies characteristic-potentially characteristic
- Potentially characteristic not implies normal-extensible automorphism-invariant
- Potentially characteristic not implies normal-potentially characteristic
- Powering-invariance does not satisfy lower central series condition in nilpotent group
- Powering-invariant and normal not implies quotient-powering-invariant
- Powering-invariant not implies divisibility-closed
- Powering-invariant not implies local powering-invariant
- Pronormal not implies join with any distinct conjugate is the whole group
- Pronormal not implies NE
- Pure subgroup of torsion-free abelian group not implies direct factor

### Q

### R

### S

- Self-normalizing not implies contranormal
- Series-equivalent characteristic subgroups may be distinct
- Strictly characteristic not implies fully invariant
- Subgroup of abelian group not implies abelian-extensible automorphism-invariant
- Subgroup of abelian group not implies abelian-potentially characteristic
- Subgroup of abelian group not implies powering-invariant
- Subhomomorph-containing not implies variety-containing
- Subisomorph-containing not implies homomorph-containing
- Subnormal not implies conjugate-permutable
- Subnormal not implies normal
- Sylow not implies CDIN
- Sylow not implies local divisibility-closed
- Sylow not implies NE