A generalization of Ramanujan's sum over finite groups

TL;DR

We address the problem of generalizing Ramanujan's sum to finite groups by analyzing the eigenvalues of the normal Cayley graph Cay(G,[x^G]), where $[x^G]$ is the set of elements with the same generated normal subgroup as $x^G$. The eigenvalues are encoded by $C_{\chi}(x) = \frac{1}{\chi(1)} \sum_{s \in [x^G]} \chi(s)$ for irreducible characters $\chi$, and the paper provides an explicit formula for $C_{\chi}(x)$ using Möbius inversion on normal subgroups, unifying Ramanujan sums with group representation theory. In the abelian case, the result reduces to the classical Ramanujan sum, giving a concrete formula in terms of group orders. The work connects spectral graph theory, representation theory, and algebraic number theory by supplying a constructive, integer-valued eigenvalue expression for integral normal Cayley graphs. Altogether, the paper extends Ramanujan's sum to finite groups and furnishes new tools for the spectral study of Cayley graphs and group-theoretic arithmetic.

Abstract

Let $G$ be a finite group, and let $x \in G$. Define $[x^G] := \{ y \in G : \langle x^G \rangle = \langle y^G \rangle \}$, where $\langle x^G \rangle$ denotes the normal subgroup of $G$ generated by the conjugacy class of $x$. In this paper, we determine an explicit formula for the eigenvalues of the normal Cayley graph $\text{Cay}(G, [x^G])$. These eigenvalues can be viewed as a generalization of classical Ramanujan's sum in the setting of finite groups. Surprisingly, the formula we derive for the eigenvalues of $\text{Cay}(G, [x^G])$ extends the known formula of classical Ramanujan's sum to the context of finite groups. This generalization not only enrich the theory of Ramanujan's sum but also provide new tools in spectral graph theory, representation theory, and algebraic number theory.

Theorems & Definitions (27)

• Theorem 1.1: babai1979spectra
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Lemma 3.2