# Central and additive group of a commutative unital ring implies potentially iterated commutator subgroup in solvable group

## Statement

Suppose is a solvable group and is a central subgroup that is isomorphic to the additive group of a commutative unital ring. Then, is a Potentially iterated commutator subgroup (?) of , i.e., there exists a group containing such that is an iterated commutator subgroup of .

In particular, is a Potentially verbal subgroup (?) and a Potentially fully invariant subgroup (?) of .

## Related facts

- Central implies finite-pi-potentially verbal in finite
- Cyclic normal implies finite-pi-potentially verbal in finite
- Homocyclic normal implies finite-pi-potentially fully invariant in finite
- Abelian normal subgroup of finite group with induced quotient action by power automorphisms is finite-pi-potentially verbal

## Proof

**Given**: A solvable group , a central subgroup of , a commutative unital ring such that is isomorphic to the additive group of .

**To prove**: There exists a group containing such that is an iterated commutator subgroup of .

**Proof**: Suppose has derived length . Consider , i.e., the group of upper-triangular matrices with s in the diagonal and entries from , under multiplication. has derived length and its center as well as the member of its derived series is a subgroup isomorphic to . Let be the central product of and with the center of identified with .

Then, the member of the derived series of equals the common subgroup . Thus, is an iterated commutator subgroup of .