Cyclotomic construction of $\lambda$-fold near-factorizations of cyclic groups

TL;DR

This paper introduces a cyclotomic construction method for creating $mbda$-fold near-factorizations of cyclic groups, specifically for primes of a certain form, extending the understanding of group factorizations.

Contribution

It provides a novel cyclotomic construction approach for $mbda$-fold near-factorizations in cyclic groups of specific prime order.

Findings

Constructs $mbda$-fold near-factorizations for primes of form $4n^4 + 12n^2 + 1$

Extends the theory of near-factorizations to new classes of cyclic groups

Offers a systematic method for such factorizations

Abstract

The study of near-factorizations of finite groups dates back to the 1950s. Recently, this topic has attracted renewed attention, and the concept has been extended to $\lambda$-fold near-factorizations, in which each non-identity group element appears exactly $\lambda \ge 1$ times. This paper presents a cyclotomic construction of $\lambda$-fold near-factorizations in the cyclic group $\mathbb{F}_p$, where $p = 4n^4 + 12n^2 + 1$ is prime for $n \ge 1$.