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

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Statement

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

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

Related facts

Proof

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

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

Proof: Suppose G has derived length l. Consider L = U(2^{l+1} - 1,R), i.e., the group of (2^l - 1) \times (2^l - 1) upper-triangular matrices with 1s in the diagonal and entries from R, under multiplication. L has derived length l + 1 and its center as well as the l^{th} member of its derived series is a subgroup isomorphic to H. Let K be the central product of G and L with the center of L identified with H.

Then, the l^{th} member of the derived series of K equals the common subgroup H. Thus, H is an iterated commutator subgroup of K.