A note on the triple product property for finite groups with abelian normal subgroups of prime index

Abstract

Three non-empty subsets $S,T,U$ of a group $G$ are said to satisfy the triple product property (TPP) if, for elements $s,s' \in S$, and $t,t' \in T$, and $u,u' \in U$, the equation $s's^{-1}t't^{-1}u'u^{-1}=1$ holds if and only if $s = s'$, $t = t'$, $u = u'$. If this is the case then $(S,T,U)$ is called a TPP triple of $G$ and $|S||T||U|$ the size of the triple. If $G$ is a finite group the triple product ratio of $G$ can be defined as the quantity $\rho(G) := \frac{\beta(G)}{|G|}$, where $\beta(G)$ is the largest size of a TPP triple of $G$, and a special case of this, the subgroup triple product ratio, is the quantity $\rho_0(G) := \frac{\beta_0(G)}{|G|}$, where $\beta_0(G)$ is the largest size of a TPP triple of $G$ composed only of subgroups. There is a conjecture that $\rho(G) \leq \frac{4}{3}$ if $G$ contains a cyclic subgroup of index $2$ \citep[Conjecture 7.6]{HM}. This note proves a more general version of this conjecture for subgroups by showing that $\rho_0(G) \leq \frac{p^2}{2p-1}$ if $G$ is any finite group that contains an abelian normal subgroup of prime index $p$, an improvement by a factor of $\frac{1}{2p-1}$ on the general upper bound of $p^2$ when $G$ contains any abelian subgroup of index $p$. In conclusion a generalised conjecture using the same upper bound is presented for $\rho$ for groups with cyclic normal subgroups of prime index, based on the known data for $\rho$ in such groups of small order.