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 finite group $G$ are said to satisfy the triple product property (TPP) if, for elements $s,s&#39; \in S$, $t,t&#39; \in T$, $u,u&#39; \in U$, the equation $s&#39;s<sup>{-1}t&#39;t<sup>{-1}u&#39;u<sup>{-1}=1$ holds if and only if $s = s&#39;$, $t = t&#39;$, $u = u&#39;$. In this case $(S,T,U)$ is called a TPP triple of $G$ and $|S||T||U|$ is called the size of the triple. The triple product ratio of $G$ can be defined as the quantity $ρ(G) := \frac{β(G)}{|G|}$, where $β(G)$ is the largest size of a TPP triple of $G$, and a special case of this, the subgroup triple product ratio, is the quantity $ρ_0(G) := \frac{β_0(G)}{|G|}$, where $β_0(G)$ is the largest size of a TPP triple of $G$ composed only of subgroups. There is a conjecture that $ρ(G) \leq \frac{4}{3}$ if $G$ contains a cyclic subgroup of index $2$. This note proves a version of this conjecture for subgroups by showing that $ρ_0(G) \leq \frac{p<sup>2}{2p-1}$ if $G$ is any group which contains an abelian normal subgroup of prime index $p$.