Exercise 18.2
Posted: Thu Mar 21, 2013 9:24 am
Exercise 18.2. Write down explicitly the character table of $D_{12}$, and show that all its entries are integers.
Use the character table to find seven distinct normal subgroups of $D_{12}. (Hint: use Proposition 17.5.)$\newcommand{\ve}{\varepsilon}\newcommand{\inv}[1]{{#1}^{-1}}$
$D_{12}=\langle a,b; a^6=a^2=1, b^{-1}ab=a^{-1}\rangle$
Máme 6 tried konjugácie (teda budeme mať 6 charakterov): $\{1\}$, $\{a^3\}$, $\{a,a^5\}$, $\{a^2,a^4\}$ $\{b,a^2b,a^4b\}$ a $\{ab,a^3b,a^5b\}$.
V tejto kapitole sme videli, ako vyzerá tabuľke charakterov dihedrálnej grupy, s.183:
Ak $\ve=e^{\frac{2\pi i}6}$, tak $\ve+\inv\ve=1$, $\ve^2+\ve^{-2}=-1$, $\ve^3=\ve^{-3}=-1$
$$
\begin{array}{c|ccccccc}
g & 1 & a^3 & a & a^2 & b & ab \\
|C_G(g)| & 12 & 12 & 6 & 6 & 4 & 4 \\\hline
\chi_1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\chi_2 & 1 & 1 & 1 & 1 & -1 & -1 \\
\chi_3 & 1 & -1 & -1 & 1 & 1 & -1 \\
\chi_4 & 1 & -1 & -1 & 1 & -1 & 1 \\
\psi_1 & 2 & -2 & 1 & -1 & 0 & 0 \\
\psi_2 & 2 & 2 &-1 & -1 & 0 & 0 \\\hline
\end{array}
$$
Normálne podgrupy vieme nájsť ako jadrá charakterov a ich prieniky.
$\ker\chi_1=G$
$\ker\chi_2=\langle a \rangle = \{1,a,\dots,a^5\}$
$\ker\chi_3=\langle a^2,b \rangle = \{1,a^2,a^4,b,a^2b,a^4b\}$
$\ker\chi_4=\langle a^2,ab \rangle = \{1,a^2,a^4,ab,a^3b,a^5b\}$
$\ker\chi_3\cap\ker\chi_4=\langle a^2\rangle = \{1,a,a^2\}$
$\ker\psi_2=\langle a^3 \rangle =\{1,a^3\}=Z(G)$
$\ker\psi_1=\{1\}$
Use the character table to find seven distinct normal subgroups of $D_{12}. (Hint: use Proposition 17.5.)$\newcommand{\ve}{\varepsilon}\newcommand{\inv}[1]{{#1}^{-1}}$
$D_{12}=\langle a,b; a^6=a^2=1, b^{-1}ab=a^{-1}\rangle$
Máme 6 tried konjugácie (teda budeme mať 6 charakterov): $\{1\}$, $\{a^3\}$, $\{a,a^5\}$, $\{a^2,a^4\}$ $\{b,a^2b,a^4b\}$ a $\{ab,a^3b,a^5b\}$.
V tejto kapitole sme videli, ako vyzerá tabuľke charakterov dihedrálnej grupy, s.183:
Ak $\ve=e^{\frac{2\pi i}6}$, tak $\ve+\inv\ve=1$, $\ve^2+\ve^{-2}=-1$, $\ve^3=\ve^{-3}=-1$
$$
\begin{array}{c|ccccccc}
g & 1 & a^3 & a & a^2 & b & ab \\
|C_G(g)| & 12 & 12 & 6 & 6 & 4 & 4 \\\hline
\chi_1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\chi_2 & 1 & 1 & 1 & 1 & -1 & -1 \\
\chi_3 & 1 & -1 & -1 & 1 & 1 & -1 \\
\chi_4 & 1 & -1 & -1 & 1 & -1 & 1 \\
\psi_1 & 2 & -2 & 1 & -1 & 0 & 0 \\
\psi_2 & 2 & 2 &-1 & -1 & 0 & 0 \\\hline
\end{array}
$$
Normálne podgrupy vieme nájsť ako jadrá charakterov a ich prieniky.
$\ker\chi_1=G$
$\ker\chi_2=\langle a \rangle = \{1,a,\dots,a^5\}$
$\ker\chi_3=\langle a^2,b \rangle = \{1,a^2,a^4,b,a^2b,a^4b\}$
$\ker\chi_4=\langle a^2,ab \rangle = \{1,a^2,a^4,ab,a^3b,a^5b\}$
$\ker\chi_3\cap\ker\chi_4=\langle a^2\rangle = \{1,a,a^2\}$
$\ker\psi_2=\langle a^3 \rangle =\{1,a^3\}=Z(G)$
$\ker\psi_1=\{1\}$