# Central implies finite-pi-potentially verbal in finite

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Central subgroup (?)) must also satisfy the second subgroup property (i.e., Finite-pi-potentally verbal subgroup (?)). In other words, every central subgroup of finite group is a finite-pi-potentally verbal subgroup of finite group.

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## Statement

Suppose is a finite group and is a central subgroup of , i.e., is contained inside the center of . Then, there exists a finite group containing such that every prime divisor of the order of also divides the order of , and is a Verbal subgroup (?) of .

## Related facts

### Weaker facts

- Abelian hereditarily normal implies finite-pi-potentially verbal in finite
- Central implies potentially verbal in finite
- Central implies potentially fully invariant in finite
- Central implies finite-pi-potentially fully invariant in finite
- Central implies finite-pi-potentially characteristic in finite

- Cyclic normal implies finite-pi-potentially verbal in finite
- Homocyclic normal implies finite-pi-potentially fully invariant in finite
- Normal not implies finite-pi-potentially characteristic in finite
- Normal not implies potentially verbal
- Normal not implies potentially fully invariant
- Central and additive group of a commutative unital ring implies potentially iterated commutator subgroup in solvable group
- NPC theorem: normal equals potentially characteristic
- Finite NPC theorem: normal equals potentially characteristic in finite groups.