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\begin{document}
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\renewcommand{\thesection}{}
\section{Algebra III - Abstraktna algebra, 1. kolokvij, 23.11.2016.}

















\begin{zad}
Dana je množica $G=\{x+y\sqrt{5}\,\mid\,x^2-5y^2=1,\,x,y\in\Q\}$. Pokaži, da je $(G,\cdot)$ grupa, kjer je $\cdot$ običajno množenje realnih števil. Ali je grupa $G$ abelska?
\end{zad}





Re.

\smallskip
$(x_1+y_1\sqrt{5})(x_2+y_2\sqrt{5})=(x_1x_2+5y_1y_2)+(x_1y_2+y_1x_2)\sqrt{5}$, $(x_1x_2+5y_1y_2)^2-5(x_1y_2+y_1x_2)^2=1$, $1=1+0\sqrt{5}\in G$, $(x+y\sqrt{5})^{-1}=x-y\sqrt{5}\in G$.

\hfill{$\square$}








\begin{zad}
Za vse podgrupe reda $8$ v grupi $\Z_{32}$ napiši vse njihove generatorje.
\end{zad}

Re.

\smallskip
$|\langle16\rangle|=2$, $|\langle8\rangle|=|\langle24\rangle|=4$, $|\langle4\rangle|=|\langle12\rangle|=|\langle20\rangle|=|\langle28\rangle|=8$.

\hfill{$\square$}






\begin{zad}
Dana je grupa $G=\{1,2,...,12\}$ z operacijo množenja po modulu 13. Določi vse homomorfizme iz grupe $G$ v grupo $(\Z_6,+)$.
\end{zad}



Re.

\smallskip
Obstaja 6 različnih homomorfizema, $\phi:G\rightarrow\Z_6$, $2^k\rightarrow ka$ za $a\in\{0,1,2,3,4,5,6\}$.

\hfill{$\square$}



\begin{zad}
Napiši vse leve odseke podgrupe $\langle(1432)\rangle$ v grupi $S_4$. 
\end{zad}



Re.

\smallskip
$\mbox{id}H$, $(12)H$, $(13)H$, $(14)H$, $(23)H$ in $(34)H$.


\hfill{$\square$}




\begin{zad}
Naj bo $G$ grupa in $a,b,c \in G$ elementi grupe $G$, za katere velja $a \cdot b \cdot c = e$, kjer je $e$ enota grupe $G$. Pokaži, da potem velja tudi $b \cdot c \cdot a=e$.
\end{zad}




Re.

\smallskip
$abc=e$, $bc=a^{-1}$, $bca=e$.


\hfill{$\square$}








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