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\begin{document}




\raggedright


\renewcommand{\thesection}{}
\section{\centerline{Algebra III - Abstraktna algebra, 12.02.2018.}}







\begin{zad}
\begin{itemize}
\item[{\rm (a)}] Naj bosta $a$ in $b$ dve celi števili, in naj bo $d=\gcd(a,b)$. Naj bo $H=\{am+bn\,|\,m,n\in\Z\}$. Pokaži, da je
$$
H=d\Z.
$$

\item[{\rm (b)}] Določi vse podgrupe grupe $(\Z,+)$.
\end{itemize}
\end{zad}






\medskip
Re. 

\smallskip
(a) $H\subseteq d\Z$: $h\in H$ $\Rightarrow$ $\exists m,n\in \Z$ t.d. $h=am+bn$, $d=\gcd(a,b)$ $\Rightarrow$ $d | a$ in $d|b$ $\Rightarrow$ $d|h$ $\Rightarrow$ $h\in d\Z$.

\smallskip
$d\Z\subseteq H$: $\ell\in d\Z$ $\Rightarrow$ ... $\Rightarrow$ $\ell\in H$.

\medskip
(b) $k\Z$ $\forall k\in\Z$.





\begin{zad}
\begin{itemize}
\item[{\rm (a)}] Pokaži, da grupi $\Z$ in $\Z\times\Z$ nista izomorfni. 

\item[{\rm (b)}] Dana je grupa $S_7$. Določi maksimalen red elementov v grupi $S_7$ (torej navečji možen red, ki ga nek element grupe $S_7$ dejansko ima). Določi število elementov reda $10$ grupe $S_7$.
\end{itemize}
\end{zad}




\medskip
Re. 

\smallskip
(a) $\Z$ je ciklična. $\Z\times\Z=\langle (1,0),(0,1) \rangle$.

\medskip
(b) $12=\lcm(3,4)=\lcm(4,3)$, $\ds{\frac{7\cdot 6\cdot 5\cdot 4\cdot 3}{5}\cdot\frac{2\cdot 1}{2}=504}$.




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[e]{{
\begin{zad}
Naj bo $\O$ grupa vseh simetrij kvadra (kuboida) (rotacija, zrcaljenje, drsno zrcaljenje,...). Grupa $\O$ deluje na množici $\{v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\}$ ogljišč kvadra. Določi stabilizator ogljišča $v_1$ glede na to delovanje. Določi stabilizator množice $\{v_1,v_8\}$ glede na to delovanje. Uporabi orbita-stabilizator izrek in izračunaj $|\O|$. Katera znana grupa je izomorfna grupi $O$?
\end{zad}
}}
{fig:text}





\medskip
Re. 

\smallskip
$|\O|=16$, $\O\cong D_4\times\Z_2$, 
$\O_1=\left\{
\left(\begin{matrix} 
1&2&3&4&5&6&7&8\\ 
1&2&3&4&5&6&7&8
\end{matrix}\right),
\left(\begin{matrix} 
1&2&3&4&5&6&7&8\\ 
1&2&6&5&4&3&7&8
\end{matrix}\right)
\right\}$, 
$\O_{\{1,8\}}=\left\{
\left(\begin{matrix} 
1&2&3&4&5&6&7&8\\ 
1&2&3&4&5&6&7&8
\end{matrix}\right),
\left(\begin{matrix} 
1&2&3&4&5&6&7&8\\ 
1&2&6&5&4&3&7&8
\end{matrix}\right)
\right\}$.






\begin{zad}
Naj bo $G$ enostavna grupa reda $1188$. Koliko elemntov reda $11$ je v $G$?
\end{zad}






\medskip
Re. 

\smallskip
$12\cdot 10=120$.















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




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