# Properties

 Label 0.1.9 Name $$\mu(9)$$ Weight $0$ Degree $1$ Real dimension $0$ Components $9$ Contained in $$\mathrm{U}(1)$$ Identity component $$\mathrm{SO}(1)$$ Component group $$C_{9}$$

## Invariants

 Weight: $0$ Degree: $1$ $\mathbb{R}$-dimension: $0$ Components: $9$ Contained in: $\mathrm{U}(1)$ Rational: no

## Identity component

 Name: $\mathrm{SO}(1)$ Index: $9$ $\mathbb{R}$-dimension: $0$ Description: $\textsf{trivial}$

## Component group

 Name: $C_{9}$ Order: $9$ Abelian: yes Generators: $\left[\zeta_{9}\right]$

## Subgroups and supergroups

 Maximal subgroups: $\mu(3)$ Minimal supergroups: $\mu(18)$, $\mu(27)$, $\mu(45)$, $\ldots$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$ $0$ $0$ $0$

## Event probabilities

 $\mathrm{Pr}[a_1=1]=\frac{1}{9}$