Non-induced modular representations of cyclic groups

TL;DR

This paper characterizes the structure of non-induced modular representations of cyclic groups over any field, revealing their rank and introducing a novel number-theoretic technique.

Contribution

It provides a complete computation of the ring of non-induced representations for cyclic groups over arbitrary fields, independent of characteristic.

Findings

The rank of the ring is $\, ext{φ}(n)$, Euler's totient function.

Introduces a 'pick-a-number' technique for expressing integers as sums of p-adic digit products.

The rank is independent of the field's characteristic.

Abstract

We compute the ring of non-induced representations for a cyclic group, $C_n$, over an arbitrary field and show that it has rank $\varphi(n)$, where $\varphi$ is Euler's totient function - independent of the characteristic of the field. Along the way, we obtain a "pick-a-number" trick; expressing an integer $n$ as a sum of products of $p$-adic digits of related integers.