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\begin{document}




\raggedright


\renewcommand{\thesection}{}
\section{\centerline{Algebra III - Abstraktna algebra, 29.11.2016.}}











\begin{zad}
\begin{itemize}
\item[{\rm(a)}]
Dani sta množici $H_0=\{\left[\begin{matrix}a&b\\c&d\end{matrix}\right]\in\Mat_{2\times 2}(\R)\,|\,a+b+c+d=0\}$ in $H_1=\{\left[\begin{matrix}a&b\\c&d\end{matrix}\right]\in\Mat_{2\times 2}(\R)\,|\,a+b+c+d=1\}$. Preveri, ali množici $H_0$ in $H_1$ tvorita grupi glede na operacijo seštevanja matrik. Odgovore utemeljite!

\item[{\rm(b)}] Naj bosta $H$ in $K$ končni podgrupi grupe $G$, ter da je $\gcd(|H|,|K|)$ praštevilo. Pokaži da je $H\cap K$ ciklična grupa.
\end{itemize}
\end{zad}

\medskip
Re.

\smallskip
(a) $H_0:$ $(a+a_1)+(b+b_1)+(c+c_1)+(d+d_1)=0$ je zaprta; je asocijativna; identiteta je $\left[\begin{matrix}0&0\\0&0\end{matrix}\right]$; $\left[\begin{matrix}-a&-b\\-c&-d\end{matrix}\right]$ je inverz za $\left[\begin{matrix}a&b\\c&d\end{matrix}\right]$; $H_1$ je grupa.

\smallskip
$H_1$ ni grupa (ni zaprta), npr.  $\left[\begin{matrix}1&0\\0&0\end{matrix}\right], \left[\begin{matrix}0&0\\0&1\end{matrix}\right]\in H_1$ ampak $\left[\begin{matrix}1&0\\0&0\end{matrix}\right]+\left[\begin{matrix}0&0\\0&1\end{matrix}\right]\not\in H_1$.


\medskip
(b) $H\cap K$ je podgrupa grupe $G$. Če je $k=|H\cap K|$ potem $k$ deli $|H|$ in $k$ deli $|K|$.

Če $k$ deli $\gcd(|H|,|K|)$ $\Rightarrow$ $k=1$ ali $k$ je praštevilo...






\begin{zad}
Naj bosta $\alpha=\left(\begin{matrix} 1&2&3&4&5&6&7&8\\2&3&1&5&4&8&6&7\end{matrix}\right)$ in $\beta=\left(\begin{matrix} 1&2&3&4&5&6&7&8\\3&1&7&8&6&5&4&2\end{matrix}\right)$. 
\begin{itemize}
\item[{\rm (a)}] Napiši $\alpha\beta$ in $\beta^2\alpha$ kot produkt disjunktnih ciklov. 
\item[{\rm (b)}] Napiši $\alpha\beta$ in $\beta^2\alpha$ kot produkt 2-ciklov (torej kot produkt transpozicij). 
\item[{\rm (c)}] Določi $\alpha^{-1}$, $\beta^{-1}$ in $(\alpha\beta)^{456}$.
\end{itemize}
\end{zad}


\medskip
Re.

\smallskip
(a) $\alpha\beta=(364758)$, $\beta^2=(178)(234)$, $\beta^2\alpha=(1376)(245)$.

\smallskip
(b) $\alpha\beta=(38)(35)(37)(34)(36)$, $\beta^2\alpha=(16)(17)(13)(25)(24)$.

\smallskip
(d) $\alpha^{-1}=(132)(45)(678)$, $\beta^{-1}=(128473)(56)$, $(\alpha\beta)^6=\id$, $(\alpha\beta)^{456}=\id$.






\begin{zad}
\begin{itemize}
\item[{\rm (a)}] Določi vse homomorfizme iz grupe $\Z_4$ v grupo $\Z_2\times\Z_2$. 
\item[{\rm (b)}] Poišči grupo permutacij, ki je izomorfna grupi $\Z_4$. Napiši Cayleyevo tabelo za $\Z_4$ ter za dobljeno  grupu permutacij. 
\end{itemize}
\end{zad}


\medskip
Re.

\smallskip
(a) Homomorfizmi so 
\begin{tabular}{| c | c  c c c |}
  \hline			
  $x$ & 0 & 1 & 2 & 3 \\
    \hline
   $\varphi_1(x)$ & (0,0) & (0,0) & (0,0) & (0,0) \\
   $\varphi_2(x)$ & (0,0) & (0,1) & (0,0) & (0,1) \\
   $\varphi_3(x)$ & (0,0) & (1,0) & (0,0) & (1,0) \\
   $\varphi_4(x)$ & (0,0) & (1,1) & (0,0) & (1,1) \\
  \hline  
\end{tabular}


\bigskip
(b) $T_0=\left(\begin{matrix}0&1&2&3\\0&1&2&3\end{matrix}\right)$, $T_1=\left(\begin{matrix}0&1&2&3\\1&2&3&0\end{matrix}\right)$, $T_2=\left(\begin{matrix}0&1&2&3\\2&3&0&1\end{matrix}\right)$, $T_3=\left(\begin{matrix}0&1&2&3\\3&0&1&2\end{matrix}\right)$...


\newpage
\begin{zad}
Naj bo $G$ dana grupa in naj bo $|G|=8$. Pokaži, da mora $G$ vsebovati element reda $2$.
\end{zad}




\medskip
Re.

\smallskip
Uporabi Lagrange-ov izrek. Če je $a\in G$ potem $|a|$ deli $|G|$. Če je $|a|=2$ potem... Če je $|a|=4$ potem $a^2\cdot a^2=e$... Če je $|a|=8$ potem $a^4\cdot a^4=e$...


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




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