On the triple product property for subgroups of finite nilpotent groups of class $2$

TL;DR

This paper investigates the triple product property (TPP) for subgroups in finite nilpotent groups of class 2, establishing upper bounds on the subgroup TPP ratio based on group structure and properties.

Contribution

It provides new bounds on the subgroup TPP ratio for finite nilpotent groups of class 2, linking it to group parameters like the center and character degrees.

Findings

  ho_0(G) < \u221a{|G:Z(G)|} for all groups of class 2

  ho_0(G)  p for p-groups with cyclic commutator subgroup of order p

  ho_0(G) = 1 for p-groups with large center or small irreducible character degrees

Abstract

A number of upper bounds are proved relating to the triple product property (TPP) for subgroups of finite nilpotent groups of class $2$. The TPP is the property defined for three non-empty subsets $S, T, U$ of a group $G$ that the group equation $s's^{-1}t't^{-1}u'u^{-1} = 1$, over pairs of elements $s', s \in S$, $t', t \in T$, $u', u \in U$, is satisfied if and only if $s' = s$, $t' = t$, $u' = u$. When $G$ is finite, and the parameter $\rho_0(G)$, called \emph{subgroup TPP ratio}, is defined as $\rho_0(G) := \max\frac{|S||T||U|}{|G|}$, where the maximum is over the collection of all triples of subgroups $S, T, U$ of $G$ satisfying the TPP, this paper proves that \textup{(1)} $\rho_0(G) < \sqrt{|G:Z(G)}$} for (all) groups of nilpotency class $2$, \textup{(2)} $\rho_0(G) \leq p$ for $p$-groups with a cyclic commutator subgroup of order $p$, \textup{(3)} $\rho_0(G) = 1$ for $p$-groups of nilpotency class $2$ with a "large" centre, loosely defined as those satisfying $p^2 \leq |G:Z(G)| \leq p^3$, \textup{(4)} and $\rho_0(G) = 1$ for $p$-groups of nilpotency class $2$ with "small" (irreducible, complex) character degrees of $1$ or $p$.