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

Three non-empty subsets $S,T,U$ of a group $G$ are said to satisfy the triple product property (TPP) if, for elements $s,s^{\mathrm{\prime }}\in S$, and $t,t^{\mathrm{\prime }}\in T$, and $u,u^{\mathrm{\prime }}\in U$, the equation $s^{\mathrm{\prime }}{s}^{-1}{t}^{\mathrm{\prime }}{t}^{-1}{u}^{\mathrm{\prime }}{u}^{-1}=1$ holds if and only if $s=s^{\mathrm{\prime }}$, $t=t^{\mathrm{\prime }}$, $u=u^{\mathrm{\prime }}$. If this is the case then $(S,T,U)$ is called a TPP triple of $G$ and $\mathrm{\mid }S\mathrm{\mid }\mathrm{\mid }T\mathrm{\mid }\mathrm{\mid }U\mathrm{\mid }$ the size of the triple. If $G$ is a finite group the triple product ratio of $G$ can be defined as the quantity $\rho (G):=\frac{\beta (G)}{\mathrm{\mid }G\mathrm{\mid }}$, 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 ${\rho }_0(G):=\frac{{\beta }_0(G)}{\mathrm{\mid }G\mathrm{\mid }}$, where ${\beta }_0(G)$ is the largest size of a TPP triple of $G$ composed only of subgroups. There is a conjecture that $\rho (G)\le \frac{4}{3}$ 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 ${\rho }_0(G)\le \frac{p^2}{2p-1}$ 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.