Skip to main content
MathOverflow

Return to Question

Language editing, and added top-level tags.
Source Link
Stefan Kohl ♦
Stefan Kohl ♦
  • 19.6k
  • 21
  • 75
  • 137

Character table Rational conjugation of elements of a finite group

SupposeLet $G$ isbe a finite group and $x, y \in G$. Two elements $x$ and $y$ of $G$ are said to be rational conjugate rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups of $G$. Assume thatLet $A$ isbe the set of all character values of $G$ and, and let $Q(A)$ is an extension$H$ be the Galois group of $Q$ in $A$. Setthe field extension $H = Gal(\dfrac{Q(A)}{Q})$$\mathbb{Q}(A)/\mathbb{Q}$. Is Is it true that the orbits of $H$ on conjugacy classes of $G$ are the same as the equivalence classes of $\sim_{r}$? Is there any proof or reference for this result?

Character table

Suppose $G$ is a finite group and $x, y \in G$. $x$ and $y$ are said to be rational conjugate $x \sim_{r} y$ if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups of $G$. Assume that $A$ is the set of all character values of $G$ and $Q(A)$ is an extension of $Q$ in $A$. Set $H = Gal(\dfrac{Q(A)}{Q})$. Is it true that the orbits of $H$ on conjugacy classes of $G$ are the same as the equivalence classes of $\sim_{r}$? Is there any proof or reference for this result?

Rational conjugation of elements of a finite group

Let $G$ be a finite group. Two elements $x$ and $y$ of $G$ are said to be rationally conjugate, written $x \sim_{r} y$, if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups of $G$. Let $A$ be the set of all character values of $G$, and let $H$ be the Galois group of the field extension $\mathbb{Q}(A)/\mathbb{Q}$. Is it true that the orbits of $H$ on conjugacy classes of $G$ are the same as the equivalence classes of $\sim_{r}$? Is there any proof or reference for this result?

used \langle and \rangle instead of < and >
Source Link
Simon Rose
Simon Rose
  • 6.3k
  • 33
  • 53

Suppose $G$ is a finite group and $x, y \in G$. $x$ and $y$ are said to be rational conjugate $x \sim_{r} y$ if and only if $<x>$$\langle x\rangle$ and $<y>$$\langle y\rangle$ are conjugate subgroups of $G$. Assume that $A$ is the set of all character values of $G$ and $Q(A)$ is an extension of $Q$ in $A$. Set $H = Gal(\dfrac{Q(A)}{Q})$. Is it true that the orbits of $H$ on conjugacy classes of $G$ are the same as the equivalence classes of $\sim_{r}$? Is there any proof or reference for this result?

Suppose $G$ is a finite group and $x, y \in G$. $x$ and $y$ are said to be rational conjugate $x \sim_{r} y$ if and only if $<x>$ and $<y>$ are conjugate subgroups of $G$. Assume that $A$ is the set of all character values of $G$ and $Q(A)$ is an extension of $Q$ in $A$. Set $H = Gal(\dfrac{Q(A)}{Q})$. Is it true that the orbits of $H$ on conjugacy classes of $G$ are the same as the equivalence classes of $\sim_{r}$? Is there any proof or reference for this result?

Suppose $G$ is a finite group and $x, y \in G$. $x$ and $y$ are said to be rational conjugate $x \sim_{r} y$ if and only if $\langle x\rangle$ and $\langle y\rangle$ are conjugate subgroups of $G$. Assume that $A$ is the set of all character values of $G$ and $Q(A)$ is an extension of $Q$ in $A$. Set $H = Gal(\dfrac{Q(A)}{Q})$. Is it true that the orbits of $H$ on conjugacy classes of $G$ are the same as the equivalence classes of $\sim_{r}$? Is there any proof or reference for this result?

Source Link
elsa haghi
elsa haghi
  • 21
  • 1

Character table

Suppose $G$ is a finite group and $x, y \in G$. $x$ and $y$ are said to be rational conjugate $x \sim_{r} y$ if and only if $<x>$ and $<y>$ are conjugate subgroups of $G$. Assume that $A$ is the set of all character values of $G$ and $Q(A)$ is an extension of $Q$ in $A$. Set $H = Gal(\dfrac{Q(A)}{Q})$. Is it true that the orbits of $H$ on conjugacy classes of $G$ are the same as the equivalence classes of $\sim_{r}$? Is there any proof or reference for this result?