The cyclic induction subgroup of the character ring

Let $G$ be a finite group. For each cyclic subgroup $C\le G$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups), induction on characters gives a homomorphism

${\mathrm{Ind}}_C^G:R(C)⟶R(G)$

(The induced character ${\mathrm{Ind}}_H^G\chi$ of a complex character, Virtual characters and the character ring $R(G)$ of a finite group).

The cyclic induction subgroup of $R(G)$ is

$I_{\mathrm{c}\mathrm{y}\mathrm{c}}(G):={\sum }_{\begin{array}{c}C\le G\\ C\text{ cyclic}\end{array}}{\mathrm{Ind}}_C^G(R(C))\subseteq R(G)\mathrm{.}$

Thus an element of $I_{\mathrm{c}\mathrm{y}\mathrm{c}}(G)$ is a finite sum

${\sum }_i{\mathrm{Ind}}_{C_i}^G({\theta }_i)$

with each $C_i\le G$ cyclic and each ${\theta }_i\in R(C_i)$.

• The subgroup $I_{\mathrm{c}\mathrm{y}\mathrm{c}}(G)$ is defined using all rational or virtual characters of cyclic subgroups, not only their trivial characters.

• The Artin relation on this page first produces $\mathrm{\mid }G\mathrm{\mid }{1}_G$ as an integral combination of permutation characters ${\mathrm{Ind}}_C^G{1}_C$, and then the theorem upgrades that relation to arbitrary elements of $R_{\mathbb{Q}}(G)$ by multiplying inside the ideal $I_{\mathrm{c}\mathrm{y}\mathrm{c}}(G)$.