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\raggedright





\renewcommand{\thesection}{}
\section{\centerline{Algebra III - Abstraktna algebra, 18.02.2016.}}







\begin{zad}
Katere od naslednjih trditev so pravilne? Odgovore utemelji!

\hspace{5mm}
(25\%)(a) Naj bodo $x_1, x_2, \ldots, x_n$ elementi poljubne grupe $G$. Potem velja 
             $$(x_1 x_2 \cdots x_n)^{-1} = x_n^{-1} \cdots x_2^{-1} x_1^{-1}.$$

\hspace{5mm}
(25\%)(b) Vsaka grupa reda $79$ je ciklična.

\hspace{5mm}
(25\%)(c) Grupa ${\mathbf Z}_{35}$ ima 24 generatorjev.

\hspace{5mm}
(25\%)(d) Grupa $G$ z enoto $e$, v kateri velja $x^2 = e$ za vsak $x \in G$, je abelska.
\end{zad}



Re.

\smallskip
(a.) $(x_1x_2...x_n)(x_yx_2...x_n)^{-1}=e$, $(x_1x_2...x_n)(x_n^{-1} \cdots x_2^{-1} x_1^{-1})=e$.

\smallskip
(b.) Naj bo $|G|=79$. Potem $\forall a\in G$, $a\ne e$, $\langle a\rangle=G$.

\smallskip
(c.) Da. ($\langle 0\rangle\ne G$, $\langle 5\rangle\ne G$, $\langle 7\rangle\ne G$,...)

\smallskip
(d.) $(xy)^2=e$, $xyxy=e$, $x^2yxy=x$, $yxy=x$, $xy=yx$.


\hfill{$\square$}








\begin{zad}
Naj bosta $\alpha=\left(\begin{matrix} 1&2&3&4&5&6&7&8\\2&3&4&5&1&7&8&6\end{matrix}\right)$ in $\beta=\left(\begin{matrix} 1&2&3&4&5&6&7&8\\1&3&8&7&6&5&2&4\end{matrix}\right)$ elementa simetrične grupe $S_8$. 

\hspace{5mm}
(40\%)(a) Napiši $\alpha$, $\beta$ in $\alpha\beta$ kot produkt disjunktnih ciklov.

\hspace{5mm}
(30\%)(b) Napiši $\alpha$, $\beta$ in $\alpha\beta$ kot produkt 2-ciklov (kot produkt transpozicij).

\hspace{5mm}
(30\%)(c) Določi $\alpha^{-2}$.
\end{zad}







Re.

\smallskip
(a.) $\alpha=(12345)(678)$, $\beta=(23847)(56)$, $\alpha\beta=(12485736)$.

\smallskip
(b.) $\alpha=(15)(14)(13)(12)(68)(67)$, $\beta=(27)(24)(28)(23)(56)$, $\alpha\beta=(16)(13)(17)(15)(18)(14)(12)$.

\smallskip
(c.) $\alpha^{-2}=(14253)(678)$.


\hfill{$\square$}









\begin{zad}
Poišči vse leve odseke podgrupe $H$ v grupi $G$, če je:

\hspace{5mm}
(40\%)(a) $G=\Z_{24}$ in $H=\langle 4\rangle$.

\hspace{5mm}
(60\%)(b) $G=S_{3}$ in $H=\langle (23)\rangle$.
\end{zad}




Re.

\smallskip
(a.) $H=\{0,4,8,12,16,20\}$, $1+H=\{1,5,9,13,17,21\}$, $2+H=\{2,6,10,14,18,22\}$, $3+H=\{3,7,11,15,19,23\}$.

\smallskip
(b.) $H=(23)H=\{\mbox{id},(23)\}$, $(12)H=(123)H=\{(12),(123)\}$, $(13)H=(132)H=\{(13),(132)\}$. 









\begin{zad}
Dana je množica $G=\{f_1,f_2,f_3,f_4\}$ kje so $f_1$, $f_2$, $f_3$ in $f_4$ preslikave definirane z
$$
f_1(x)=x,\qquad f_2(x)=-x,\qquad f_3(x)=\frac{1}{x},\qquad f_4(x)=-\frac{1}{x},\qquad x\in\R, x\not=0.
$$
Pokaži, da je $(G,\circ)$ grupa, kjer je $\circ$ označuje običajno komponiranje funkcij. 
\end{zad}






Re.

\smallskip
\hspace{25mm}
{\begin{tabular}{c|cccccccc|}
$\circ$&$f_1$&$f_2$&$f_3$&$f_4$\\\midrule
$f_1$&$f_1$&$f_2$&$f_3$&$f_4$\\
$f_2$&$f_2$&$f_1$&$f_4$&$f_3$\\
$f_3$&$f_3$&$f_4$&$f_1$&$f_2$\\
$f_4$&$f_4$&$f_3$&$f_2$&$f_1$\\\midrule
\end{tabular}}


\hfill{$\square$}









\vfill{Naloge so povzeti z
\url{http://osebje.famnit.upr.si/~penjic/}
}
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