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

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 $s^{\prime }{s}^{-1}{t}^{\prime }{t}^{-1}{u}^{\prime }{u}^{-1}=1$, over pairs of elements $s^{\prime },s\in S$, $t^{\prime },t\in T$, $u^{\prime },u\in U$, is satisfied if and only if $s^{\prime }=s$, $t^{\prime }=t$, $u^{\prime }=u$. 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)} ${\rho }_0(G)\le p$ 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 $p^2\le |G:Z(G)|\le p^3$, \textup{(4)} and ${\rho }_0(G)=1$ for $p$-groups of nilpotency class $2$ with "small" (irreducible, complex) character degrees of $1$ or $p$.

17 pages, including abstract and references