On the triple product property for subgroups of finite nilpotent groups of class $2$
Sandeep R. Murthy
TL;DR
This work analyzes the subgroup triple product property (TPP) in finite groups, focusing on the subgroup TPP ratio $\rho_0(G)$. It proves that for all groups of nilpotency class $2$, $\rho_0(G)<\sqrt{|G:Z(G)|}$, and strengthens this bound to $\le p$ for $p$-groups with a cyclic commutator subgroup of order $p$. Furthermore, it shows $\rho_0(G)=1$ for $p$-groups of class $2$ with either a large centre ($p^2\le|G:Z(G)|\le p^3$) or with small complex character degrees ($\mathrm{cd}(G)=\{1,p\}$). The results are motivated by computational data and have potential implications for the matrix-multiplication context through subgroup realizations, using quotients and central/commutator structure to bound the TPP capacity.
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 $ρ_0(G)$, called \emph{subgroup TPP ratio}, is defined as $ρ_0(G) := \max\frac{|S||T||U|}{|G|}$, 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)} $ρ_0(G) < \sqrt{|G:Z(G)}$} for (all) groups of nilpotency class $2$, \textup{(2)} $ρ_0(G) \leq p$ for $p$-groups with a cyclic commutator subgroup of order $p$, \textup{(3)} $ρ_0(G) = 1$ 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 $ρ_0(G) = 1$ for $p$-groups of nilpotency class $2$ with "small" (irreducible, complex) character degrees of $1$ or $p$.