FZ implies finite derived subgroup

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
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This result was proved by Schur and is sometimes termed the Schur-Baer theorem.

Statement

Verbal statement

If the inner automorphism group (viz the quotient by the center) of a group is finite, so is the commutator subgroup. In fact, there is an explicit bound on the size of the commutator subgroup as a function of the size of the inner automorphism group.

Symbolic statement

Let $G$ be a group such that $Inn(G)=G/Z(G)$ is finite. Then, $G^{\prime }=[G,G]$ is also finite. In fact, if $|G/Z(G)|=n$, then $G^{\prime }$ has size at most $n^{2n^3}$.

Property-theoretic statement

The group property of being a FZ-group (viz having a finite inner automorphism group) implies the group property of being commutator-finite viz having a finite commutator subgroup.

Proof

The proof involves two steps:

• Showing that the number of distinct commutators is at most $n^2$: This follows from the fact that the commutator $[x,y]$ depends only on the quotients of $x$ and $y$ modulo $Z(G)$, and thus there are $n^2$ possibilities.
• Showing that for any element in the commutator subgroup, there is a minimal word for that element with each commutator occuring at most $n$ times: This shows that any element of the commutator subgroup has a word in terms of the commutators, of length at most $n^3$, and this completes the proof.

Converses

The direct converse of the Schur-Baer theorem is false, but there are the following partial converses:

• For any commutator set-finite group (group with only finitely many commutators), the quotient by the second term of its upper central series, is a finite group; moreover, the order of this quotient is bounded from above by a function of the size of the set of commutators.
• Any finitely generated group which also has finitely many commutators, is a FZ-group (viz., its inner automorphism group is finite). Moreover, the size of the inner automorphism group is bounded by a function of the size of a generating set, and the number of commutators.