Exercise 21.5 - opäť 21-prvková podgrupa v $S_7$

[Martin Sleziak]

Exercise 21.5. Let $G=S_7$ and let $H=\langle a,b \rangle$, where
$$a=(1234567), b=(235)(476)$$
as in Example 21.25. Let $\phi$ and $\psi$ be the irreducible characters of $H$ which are given by
$$\begin{array}{c|ccccc}\hline
g_i & 1 & a & a^3 & b & b^2 \\
|C_H(g_i)| & 21 & 7 & 7 & 3 & 3 \\\hline
\phi & 1 & 1 & 1 & 1 & 1 \\
\psi & 3 & \eta+\eta^2+\eta^4 & \eta^3+\eta^5+\eta^6 & 0 & 0 \\\hline
\end{array}$$
where $\eta=e^{2\pi i/7}$ (see Example 21.25).

You are given that $|C_G(a)|=7$ and $|C_G(b)|=18$. Calculate the values of the induced characters $\phi\uparrow G$ and $\psi\uparrow G$.

Ak $\chi$ je charakter grupy $H$, tak dostaneme
$\chi\uparrow G(1)=240\chi(1)$ lebo $|G|/|H|=7!/21=6\cdot 5 \cdot 4 \cdot 2=240$
$\chi\uparrow G(a)=\chi(a)+\chi(a^3)$, lebo $|C_G(a)|=|C_H(a)|=|C_H(a^3)|=7$
$\chi\uparrow G(b)=6(\chi(b)+\chi(b^2))$, lebo $|C_G(b)|=18$ a $|C_H(b)|=|C_H(b^2)|=3$

Pre ostatné triedy konjugácie budú indukované charaktery nulové, stačí sa pozerať na $1$, $a$, $b$.
$$\begin{array}{c|ccc}\hline
& 1 & a & b \\\hline
\phi\uparrow G & 240 & 2 & 12\\
\psi\uparrow G & 720 &-1 & 0\\\hline
\end{array}$$

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Môžeme si ešte všimnúť, že pre $\alpha=\eta+\eta^2+\eta^4$ a $\overline\alpha=\eta^3+\eta^5+\eta^6$ platí
$\alpha+\overline\alpha=-1$
$\alpha\cdot\overline\alpha=(\eta+\eta^2+\eta^4)(\eta^3+\eta^5+\eta^6)=3+\eta+\eta^2+\eta^3+\eta^4+\eta^5+\eta^6=2$
Teda sú to tie isté čísla, ktoré sme dostali v Exercise 17.2, len vyjadrené iným spôsobom.