# Category:Group property non-implications

From Groupprops

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

View other property non-implication categories

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

The following 70 pages are in this category, out of 70 total. The count *includes* redirect pages that have been included in the category. Redirect pages are shown in italics.

### A

### C

### F

- FC not implies BFC
- FC not implies finite derived subgroup
- Finite derived subgroup not implies FZ
- Finite not implies composition factor-permutable
- Finite not implies composition factor-unique
- Finite solvable not implies subgroups of all orders dividing the group order
- Finite solvable not implies supersolvable
- Finitely generated and parafree not implies free
- Finitely generated and solvable not implies finitely presented
- Finitely generated and solvable not implies polycyclic
- Finitely generated not implies finitely presented
- Finitely generated not implies Noetherian
- Finitely generated not implies residually finite
- Finitely presented and solvable not implies polycyclic
- Finitely presented not implies Noetherian

### G

### L

### N

- Nilpotent automorphism group not implies abelian automorphism group
- Nilpotent not implies abelian
- Nilpotent not implies ACIC
- Nilpotent not implies generated by abelian normal subgroups
- Nilpotent not implies nilpotent automorphism group
- Nilpotent not implies UL-equivalent
- No proper nontrivial transitively normal subgroup not implies simple
- Noetherian not implies finitely presented
- Normalizer condition not implies nilpotent

### P

### R

### S

- Same Hall-Senior genus not implies character table-equivalent
- Simple not implies co-Hopfian
- Solvable and generated by finitely many periodic elements not implies periodic
- Solvable not implies nilpotent
- Solvable not implies solvable automorphism group
- Subgroups of all orders dividing the group order not implies supersolvable
- Subgroups of all orders dividing the group order not implies Sylow tower
- Supersolvable not implies nilpotent
- Sylow tower not implies subgroups of all orders dividing the group order