[Paper Review] Commutator Expansions and the Schur Multiplier

Ammu E Antony, Komma Patali+1 more|arXiv (Cornell University)|Jun 23, 2019

Homotopy and Cohomology in Algebraic Topology17 references4 citations

TL;DR

This paper establishes that for odd prime $ p $, the exponent of the second homology group $ H_2(G,\mathbb{Z}) $ of a $ p $-group $ G $ of class $ p $ divides the exponent of $ G $. It further proves a bound on the exponent of $ H_2(G,\mathbb{Z}) $ in terms of $ \e(G)^n $, where $ n = 1 + \lceil \log_{p-1} \frac{c+1}{p+1} \rceil $, using commutator expansions and Schur multiplier techniques.

ABSTRACT

We prove that the exponent of the second homology group $H_2(G,\mathbb{Z})$ divides the exponent of $G$ for $p$-groups of class $p$, where $p$ is an odd prime. Moreover, we prove that $\e(H_2(G,\mathbb{Z}))\mid (\e(G))^n$, where $n = 1+\ceil{\log_{p-1} \frac{c+1}{p+1}}$.

Motivation & Objective

• To determine the exponent of the second homology group $ H_2(G,\mathbb{Z}) $ for $ p $-groups of class $ p $, where $ p $ is an odd prime.
• To establish a quantitative relationship between the exponent of $ H_2(G,\mathbb{Z}) $ and the exponent of the group $ G $.
• To extend known results on the Schur multiplier by incorporating commutator expansion techniques in the context of $ p $-groups.

Proposed method

• Utilizes commutator expansions to analyze the structure of $ p $-groups of class $ p $.
• Applies techniques from the theory of Schur multipliers to bound the exponent of $ H_2(G,\mathbb{Z}) $.
• Employs logarithmic bounds involving $ \log_{p-1} \frac{c+1}{p+1} $ to derive the exponent multiplier $ n $.
• Derives the exponent bound $ \e(H_2(G,\mathbb{Z})) \mid \e(G)^n $ using group-theoretic and cohomological arguments.
• Relies on the class $ c $ of the group $ G $ to determine the growth rate of the exponent multiplier $ n $.

Experimental results

Research questions

• RQ1 What is the exponent of $ H_2(G,\mathbb{Z}) $ for a $ p $-group $ G $ of class $ p $, where $ p $ is an odd prime?
• RQ2 How does the exponent of $ H_2(G,\mathbb{Z}) $ relate to the exponent of $ G $ in such groups?
• RQ3 Can commutator expansions be used to derive effective bounds on the Schur multiplier in $ p $-groups of class $ p $?
• RQ4 What is the precise dependence of the exponent of $ H_2(G,\mathbb{Z}) $ on the group exponent and class $ c $?

Key findings

• The exponent of $ H_2(G,\mathbb{Z}) $ divides the exponent of $ G $ for $ p $-groups of class $ p $, where $ p $ is an odd prime.
• A bound is established: $ \e(H_2(G,\mathbb{Z})) \mid \e(G)^n $, with $ n = 1 + \lceil \log_{p-1} \frac{c+1}{p+1} \rceil $.
• The exponent of the Schur multiplier is shown to grow at most polynomially in terms of $ \e(G) $, with the exponent $ n $ depending on the group's class $ c $ and prime $ p $.
• The bound is effective and explicitly quantifies the relationship between group exponent and Schur multiplier exponent.
• The result generalizes known exponent bounds for $ H_2(G,\mathbb{Z}) $ in the context of $ p $-groups of class $ p $.