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\def\O{{\mbox{$\mathcal{O}$}}}

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\def\diag{{\mbox{\rm diag}}}
\def\Mat{{\mbox{\rm Mat}}}
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\begin{document}




\raggedright


\renewcommand{\thesection}{}
\section{\centerline{Algebra III - Abstraktna algebra, 22.01.2018.}}







\begin{zad}
\begin{itemize}
\item[{\rm (a)}] Naj bo dana grupa $G=\Mat_{2\times 2}(\R)=\left\{
\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\,\Big|\,a,b,c,d\in\R
\right\}$ z operacijo seštevanja po komponentah. Pokaži, da je množica
$
H=\{A\in G\,|\,\mbox{sled}(A)=0\}
$
podgrupa grupe $G$.

\item[{\rm (b)}] Naj bo $H$ prava podgrupa grupe $G$ (to je, $H\le G$ in $H\ne G$). Pokaži, da je $\langle G\backslash H\rangle=G$.
\end{itemize}
\end{zad}






\medskip
Re. 

\smallskip
(a) $B=\left(\begin{matrix}-1&2\\3&1\end{matrix}\right)$ $\Rightarrow$ $\trag(B)=0$ $\Rightarrow$ $H\ne\emptyset$.

\smallskip
$A=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)$, 
$B=\left(\begin{matrix}x&y\\z&w\end{matrix}\right)$,
$A,B\in H$ $\Rightarrow$ $a+d=0$, $x+w=0$ $\Rightarrow$ $a+x+d+w=0$ $\Rightarrow$ $A+B\in H$.

\smallskip
$-A=\left(\begin{matrix}-a&-b\\-c&-d\end{matrix}\right)$. 

\medskip
(b) $G=\{e,a_1,...,a_k,a_{k+1},...\}$, $H=\{e,a_1,...,a_k\}$ $\Rightarrow$ $G\backslash H=\{a_{k+1},a_{k+2},...\}$ ($G\backslash H\subseteq G$). Če $\langle G\backslash H\rangle\ne G$ potem $\exists b\in G$ t.d. $b\not\in\langle G\backslash H\rangle$.

\smallskip
$1^\circ$ $b\in H$. $\exists c\not\in H$ t.d. $cH\ne H$ $\Rightarrow$ $cH\subseteq G\backslash H$ ($cH\cap H=\emptyset$) $\Rightarrow$ $\exists d\in G\backslash H$, $cb=d$ $\Rightarrow$ $b=c^{-1}d\in G\backslash H$ $\Rightarrow$ $b\not\in H$, protislovje.

\smallskip
$2^\circ$ $b\not\in H$ $\Rightarrow$ $b\in G\backslash H$, protislovje.





\begin{zad}
Naj bo 
$$
\pi=
\left(\begin{matrix} 
1&2&3&4&5&6&7&8\\ 
2&4&6&7&5&1&3&8
\end{matrix}\right)
\left(\begin{matrix} 
1&2&3&4&5&6&7&8\\ 
8&7&3&5&1&2&6&4
\end{matrix}\right).
$$
Izračunaj $\pi^{-1}$ in $\pi^{2018}$.
\end{zad}




\medskip
Re. 

\smallskip
$|\pi|=15$, $\pi^{2018}=(178)(24356)$,
$\pi^{-1}=
\left(\begin{matrix} 
1&2&3&4&5&6&7&8\\ 
7&5&2&6&4&3&8&1
\end{matrix}\right)$.




\begin{zad}
\begin{itemize}
\item[{\rm (a)}] Naj bo $G=(\Mat_{2\times 2}(\Z),+)$ in $H=\{A\in G\,|\,\mbox{sled}(A)=0\}$. Pokaži, da je $H$ podgrupa edinka v $G$ in da je $G/H\cong\Z$.
\item[{\rm (b)}] Konstruiraj Cayley-evo tabelo za $\Aut(\Z_8)$.
\end{itemize}
\end{zad}



\medskip
Re. 

\smallskip
(a) $\phi:G\rightarrow\Z$, $\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\longrightarrow a+d$, $\phi$ je homomorfizem, $\phi(G)=\Z$, $\ker(\phi)=H$.

\bigskip
(b) $f_1(1)=1$, $f_3(1)=3$, $f_5(1)=5$, $f_7(1)=7$.

\medskip
{\footnotesize\begin{tabular}{c|cccc|}
$\circ$ & $f_1$ & $f_2$ & $f_3$ & $f_4$ \\\midrule
$f_1$ & $f_1$ & $f_3$ & $f_5$ & $f_7$ \\ 
$f_3$ & $f_3$ & $f_1$ & $f_7$ & $f_5$ \\ 
$f_4$ & $f_5$ & $f_7$ & $f_1$ & $f_3$ \\ 
$f_5$ & $f_7$ & $f_5$ & $f_3$ & $f_1$ \\ \midrule
\end{tabular}}






\begin{zad}
Center grupe $G$ je definiran na naslednji način: $Z(G)=\{a\in G\,|\,ax=xa\,\mbox{ za }\,\forall x\in G\}$. Pokaži, da je 
$$
Z(\GL_2(\R))=\R\cdot\id.
$$
\end{zad}




\medskip
Re. 

\smallskip
$X=\left(\begin{matrix}0&1\\1&0\end{matrix}\right)$ $\Rightarrow$ $X\in\GL_2(\R)$.

\smallskip
$A=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)$.

\smallskip
$AX=XA$ $\Rightarrow$ $a=d$, $b=c$.

\smallskip
$Y=\left(\begin{matrix}1&1\\1&0\end{matrix}\right)$ $\Rightarrow$ $Y\in\GL_2(\R)$.

\smallskip
$AY=YA$ $\Rightarrow$ $b=0$.











\vfill{Več na
\url{http://osebje.famnit.upr.si/~penjic/}
}




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