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

TL;DR

This paper investigates the triple product property in finite groups, proving an upper bound for the subgroup triple product ratio in groups with abelian normal subgroups of prime index, and proposes a generalized conjecture for broader classes.

Contribution

It establishes a tighter upper bound for the subgroup triple product ratio in groups with abelian normal subgroups of prime index, extending previous conjectures.

Findings

Proves  ho_0(G)  rac{p^2}{2p-1} for groups with abelian normal subgroups of prime index p.

Improves the known upper bound for the subgroup triple product ratio by a factor of  ho_0(G)  p^2.

Suggests a generalized conjecture for the triple product ratio  ho(G) in groups with cyclic 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.