Transversal Difference Numbers in Finite Abelian Quotients

Given \(H\leq G\) finite abelian groups, a transversal \(T\subseteq G\) for \(G/H\) has fixed size \(|G/H|\), but its ambient difference support \(D(T)=T-T\) can vary with the embedding of \(H\) in \(G\).

We call $ \delta(G,H)=\min_T |D(T)| $ the transversal difference number of the pair \((G,H)\).

This invariant is related to finite abelian factorisation, tiling complements, and small-sumset questions, and is motivated by recent work regarding ambient Galois labels in CRT transforms for cyclotomic-subfield homomorphic encryption.

We prove various results regarding this invariant, including a general lower bound $\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 \(H\).

The bound is sharp for cyclic quotients, and Kneser's theorem gives a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors.

These results isolate the first genuinely new residual obstruction, namely the same-prime square plane \[ G=(\mathbb Z/p^2\mathbb Z)^2,\qquad H=pG. \] For odd \(p\), this case is the technical core of the paper.

Here transversals are graphs of functions \(\mathbb F_p^2\to \mathbb F_p^2\), and \(D(T)\) decomposes into carry-corrected finite-field derivative images.

We conjecture that \[ \delta(G,H)=(2p-1)^2 \] for all odd primes \(p\), prove the unconditional lower bound \(3p^2-p-1\), and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture.