Number of equivalence classes under rational conjuacy

This article defines an arithmetic function on groups
View other such arithmetic functions

Definition

Let $G$ be a group. The number of equivalence classes under rational conjugacy, also termed the number of conjugacy classes of cyclic subgroups, is defined as follows:

1. It is the number of equivalence classes of elements in $G$ under the equivalence relation: $x\sim y\iff \langle x\rangle$ is conjugate to $\langle y\rangle$.
2. It is the number of equivalence classes of cyclic subgroups of $G$ under the equivalence relation of being conjugate subgroups.
3. (For a finite group): It is the number of representations of $G$ over a splitting field of characteristic zero that have rational-valued characters and that have no proper nonzero subrepresentation with rational-valued character.
4. (For a finite group): It is the number of irreducible representations of $G$ over $\mathbb {Q}$, the field of rational numbers. Here, irreducible simply means irreducible over $\mathbb {Q}$, not necessarily absolutely irreducible.

Equivalence of definitions

Further information: number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy

Relation with other arithmetic functions

Bigger arithmetic functions

Function	Why it's bigger (subset or quotient)?	Case of equality (when both numbers are finite)
number of conjugacy classes	quotient (viewed as equivalence relations on elements of the group)	rational group
number of conjugacy classes of subgroups	subset (picks out only those conjugacy classes that are of cyclic subgroups)	cyclic group

Smaller arithmetic functions

Function	Why it's smaller (subset or quotient)?	Case of equality (when both numbers are finite)
number of conjugacy classes of rational elements	subset	rational group