Overview
The study investigates an invariant denoted as $\delta(G,H)$, termed the transversal difference number, for pairs of finite abelian groups $(G,H)$, where $H$ is a subgroup of $G$. This invariant is defined as $\min_T |D(T)|$, where $T$ is a transversal for $G/H$, and $D(T) = T-T$ represents its ambient difference support. While the size of a transversal $T$ for $G/H$ is fixed at $|G/H|$, the ambient difference support $D(T)$ can vary depending on the embedding of $H$ within $G$.
Research Context
The transversal difference number $\delta(G,H)$ is related to concepts in finite abelian factorization, tiling complements, and small-sumset questions. Its motivation stems from recent work concerning ambient Galois labels in Chinese Remainder Theorem (CRT) transforms for cyclotomic-subfield homomorphic encryption. This establishes a connection between a theoretical invariant in group theory and applications in cryptographic methods.
Approach
The research establishes various results pertaining to the transversal difference number $\delta(G,H)$. A general lower bound is derived: $\delta(G,H)\geq 2|G/H|-m(G,H)$, where $m(G,H)$ represents the largest order of a subgroup of $G$ that is disjoint from $H$. The sharpness of this bound is demonstrated for cyclic quotients.
Kneser's theorem is leveraged to provide a cross-transversal estimate. This estimate leads to the identification of exact product families characterized by one nonsplit cyclic coordinate and arbitrary split factors. These findings contribute to isolating a specific type of residual obstruction.
A particular focus is placed on the same-prime square plane, which is defined by $G=(\mathbb Z/p^2\mathbb Z)^2$ and $H=pG$. For odd primes $p$, this scenario constitutes the technical core of the paper.
Findings
- A general lower bound for the transversal difference number is $\delta(G,H)\geq 2|G/H|-m(G,H)$, where $m(G,H)$ is the largest order of a subgroup of $G$ disjoint from $H$.
- This lower bound is sharp for cyclic quotients.
- Kneser's theorem provides a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors.
- The research isolates the same-prime square plane $G=(\mathbb Z/p^2\mathbb Z)^2$, $H=pG$ as a genuinely new residual obstruction.
- For the case of $G=(\mathbb Z/p^2\mathbb Z)^2$ and $H=pG$ (where $p$ is an odd prime), transversals are described as graphs of functions $\mathbb F_p^2\to \mathbb F_p^2$.
- In this specific case, $D(T)$ decomposes into carry-corrected finite-field derivative images.
- For odd primes $p$ in the same-prime square plane case, an unconditional lower bound of $3p^2-p-1$ for $\delta(G,H)$ is proven.
- Evidence for the conjecture $\delta(G,H)=(2p-1)^2$ for all odd primes $p$ is provided through small-prime calculations, probabilistic arguments, and fixed-polynomial analysis.
Why This Matters
The invariant $\delta(G,H)$ has implications for finite abelian factorization, tiling complements, and small-sumset questions. Its relevance is underscored by its connection to ambient Galois labels used in CRT transforms for cyclotomic-subfield homomorphic encryption, suggesting a potential impact on the theoretical underpinnings of certain cryptographic schemes.