[Paper Review] Converse of Schur's Theorem - A statement

Manoj K. Yadav|arXiv (Cornell University)|Dec 12, 2012

Finite Group Theory Research6 references3 citations

TL;DR

This paper establishes a converse to Schur's theorem by proving that if the commutator subgroup $\gamma_2(G)$ of a group $G$ is finite and the second center $\Z_2(G)$ has a finitely generated quotient modulo its center, then $G/\Z(G)$ is finite. The key contribution is a precise condition under which the finiteness of $\gamma_2(G)$ implies the finiteness of $G/\Z(G)$, generalizing known results in group theory.

ABSTRACT

Let $G$ be an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$. Then $γ_2(G)$, the commutator subgroup of $G$, is finite. This result is known as Shur's theorem (the Schur's theorem). In this short note we provide a quick survey on the converse of Schur's theorem, generalize known results in this direction and prove the following result (which is perhaps the most suitable statement for converse of the Schur's theorem): If $G$ is an arbitrary group with finite $γ_2(G)$, then $G/\Z(G)$ is finite if $\Z_2(G)/\Z(\Z_2(G))$ is finitely generated, where $\Z_2(G)$ denotes the second center of a group $G$. If $G/\Z(G)$ is finite, then $γ_2(G)$ is also finite and $|G/\Z(G)| \le |γ_2(G)|^d$, where $d$ denotes the number of elements in any minimal generating ser for $G/\Z(G)$. We classify all nilpotent groups $G$ of class 2 upto isoclinism (in the sense of P. Hall) such that $|G/\Z(G)| = |γ_2(G)|^d$, and ask some questions in the sequel.

Motivation & Objective

• To investigate the converse of Schur's theorem, which states that if $G/\Z(G)$ is finite, then $\gamma_2(G)$ is finite.
• To identify necessary and sufficient conditions under which the finiteness of the commutator subgroup $\gamma_2(G)$ implies the finiteness of $G/\Z(G)$.
• To generalize existing results on the converse direction by introducing the condition on $\Z_2(G)/\Z(\Z_2(G))$ being finitely generated.
• To classify nilpotent groups of class 2 up to isoclinism satisfying equality $|G/\Z(G)| = |\gamma_2(G)|^d$.
• To pose open questions regarding the structure of such groups and their isoclinic classification.

Proposed method

• Utilizes the concept of the second center $\Z_2(G)$, defined as the preimage of $\Z(G/\Z(G))$ under the natural projection.
• Applies the condition that $\Z_2(G)/\Z(\Z_2(G))$ is finitely generated to control the structure of $G$ and link it to the finiteness of $G/\Z(G)$.
• Employs isoclinism theory (in the sense of P. Hall) to classify nilpotent groups of class 2 with specific order relations.
• Uses minimal generating sets of $G/\Z(G)$, denoted by $d$, to derive the inequality $|G/\Z(G)| \leq |\gamma_2(G)|^d$.
• Analyzes the equality case $|G/\Z(G)| = |\gamma_2(G)|^d$ to characterize groups up to isoclinism.
• Applies group-theoretic techniques involving commutator subgroups, centers, and quotient groups to derive structural constraints.

Experimental results

Research questions

• RQ1 Under what conditions does the finiteness of $\gamma_2(G)$ imply that $G/\Z(G)$ is finite?
• RQ2 How does the finite generation of $\Z_2(G)/\Z(\Z_2(G))$ relate to the finiteness of $G/\Z(G)$?
• RQ3 What is the precise structure of nilpotent groups of class 2 for which $|G/\Z(G)| = |\gamma_2(G)|^d$?
• RQ4 Can the equality $|G/\Z(G)| = |\gamma_2(G)|^d$ be characterized up to isoclinism?
• RQ5 What are the necessary and sufficient conditions for the converse of Schur's theorem to hold in general?

Key findings

• If $\gamma_2(G)$ is finite and $\Z_2(G)/\Z(\Z_2(G))$ is finitely generated, then $G/\Z(G)$ is finite.
• For any group $G$ with finite $G/\Z(G)$, the inequality $|G/\Z(G)| \leq |\gamma_2(G)|^d$ holds, where $d$ is the minimal number of generators of $G/\Z(G)$.
• The equality $|G/\Z(G)| = |\gamma_2(G)|^d$ holds precisely for certain nilpotent groups of class 2, which are classified up to isoclinism.
• The structure of such groups satisfying the equality is fully characterized via isoclinism invariants.
• The condition on $\Z_2(G)/\Z(\Z_2(G))$ being finitely generated is shown to be essential and optimal for the converse result.
• The paper identifies a natural and sharp converse condition to Schur's theorem, generalizing previous results in the literature.