\documentclass[12pt,a4paper]{article}
\pagestyle{empty}

%%\usepackage[cp1250]{inputenc} 
\usepackage[cp1250]{inputenc} 
\usepackage[left=16mm,right=17mm,top=15mm,bottom=10mm]{geometry}
\usepackage{makeidx, graphicx, amsmath, amssymb, amsthm, ae}
\usepackage[croatian, english]{babel}
\usepackage{esint, hvfloat}
%%\usepackage{systeme,mathtools}
\usepackage{booktabs, multicol, color}




\def\R{{\mbox{$\mathbb R$}}}
\def\NN{{\mbox{$\mathbb N$}}}
\def\Z{{\mbox{$\mathbb Z$}}}
\def\Q{{\mbox{$\mathbb Q$}}}
\def\F{{\mbox{$\mathbb F$}}}
\def\C{{\mbox{$\mathbb C$}}}
\def\E{{\mbox{$\mathcal{E}$}}}
\def\M{{\mbox{$\mathcal{M}$}}}
\def\N{{\mbox{$\mathcal{N}$}}}
\def\S{{\mbox{$\mathcal{S}$}}}
\def\L{{\mbox{$\mathcal{L}$}}}
\def\P{{\mbox{$\mathcal{P}$}}}
\def\B{{\mbox{$\mathcal{B}$}}}
\def\X{{\mbox{$\mathcal{X}$}}}
\def\V{{\mbox{$\mathcal{V}$}}}
\def\Y{{\mbox{$\mathcal{Y}$}}}
\def\W{{\mbox{$\mathcal{W}$}}}
\def\dist{{\mbox{\rm dist}}}
\def\span{{\mbox{\rm span}}}
\def\ecc{{\mbox{\rm ecc}}}
\def\max{{\mbox{\rm max}}}
\def\spec{{\mbox{\rm spec}}}
\def\rank{{\mbox{\rm rank}}}
\def\mod{{\mbox{\rm mod}}}
\def\div{{\mbox{\rm div}}}
\def\trag{{\mbox{\rm trag}}}
\def\tr{{\mbox{\rm tr}}}
\def\dim{{\mbox{\rm dim}}}
\def\diag{{\mbox{\rm diag}}}
\def\Mat{{\mbox{\rm Mat}}}
\def\ker{{\mbox{\rm ker}}}
\def\im{{\mbox{\rm im}}}
\def\gcd{{\mbox{\rm gcd}}}
\def\A{{\mbox{$\pmb{A}$}}}
\def\J{{\mbox{$\pmb{J}$}}}
\def\I{{\mbox{$\pmb{I}$}}}
\def\j{{\mbox{$\pmb{j}$}}}
\def\0{{\mbox{\boldmath{$0$}}}}
\def\algmult{{\mbox{\rm alg\,mult}}}
\def\geomult{{\mbox{\rm geo\,mult}}}
\def\char{{\mbox{\rm char}}}
\def\du{{\mbox{\rm \,d$u$}}}
\def\dx{{\mbox{\rm \,d$x$}}}
\def\dt{{\mbox{\rm \,d$t$}}}
\def\dy{{\mbox{\rm \,d$y$}}}
\def\dz{{\mbox{\rm \,d$z$}}}
\def\tg{{\mbox{\rm tg\,}}}
\def\arctg{{\mbox{\rm arc\,tg\,}}}
\def\arcsin{{\mbox{\rm arc\,sin\,}}}
\def\ctg{{\mbox{\rm ctg\,}}}



\newtheoremstyle{zaFiguru} %name
  {5mm}    %Space above, empty = `usual value'
  {5mm}    %Space below
  {\rm}   %Body font
  {}     %Indent amount (empty = no indent, \parindent = para indent)
  {\bf}   %Thm head font
  {}      %Punctuation after thm head
  {2mm}   %Space after thm head: \newline = linebreak
  {\thmname{#1} \thmnumber{\Large #2.}\thmnote{\,(#3)}}       %Thm head spec  
%%\swapnumbers
\theoremstyle{zaFiguru}
\newtheorem{zad}{}


\setlength{\textheight}{260mm}
\setlength{\textwidth}{180mm}
\begin{document}




\raggedright





\renewcommand{\thesection}{}
\section{\centerline{Algebra III - Abstraktna algebra, 30.08.2016.}}
















\begin{zad}
Naj bo $G$ cikli\v{c}na grupa generirana z matriko
$$
A=\left[
\begin{matrix}
1&1\\
-1&0
\end{matrix}
\right].
$$

\smallskip
\hspace{5mm}
(30\%)(a) Kateri so preostali elementi grupe $G$? Svojo trditev obrazlo\v{z}ite.

\smallskip
\hspace{5mm}
(30\%)(b) Zapi\v{s}i Cayley-evo tabelo za grupo $G$.

\smallskip
\hspace{5mm}
(10\%)(c) Ali je $G$ abelska grupa?

\smallskip
\hspace{5mm}
(30\%)(c) Napi\v{s}ite vse generatorje grupe $G$.
\end{zad}



Re.

\smallskip
(a) $A=\left[\begin{matrix}1&1\\-1&0\end{matrix}\right]$, $A^2=\left[\begin{matrix}0&1\\-1&-1\end{matrix}\right]$, $A^3=\left[\begin{matrix}-1&0\\0&-1\end{matrix}\right]$, $A^4=\left[\begin{matrix}-1&-1\\1&0\end{matrix}\right]$, $A^5=\left[\begin{matrix}0&-1\\1&1\end{matrix}\right]$, $A^6=\left[\begin{matrix}1&0\\0&1\end{matrix}\right]=I$.

\smallskip
(b) 
${\it\begin{tabular}{c|cccccccc|}
\, & $I$ & $A$ & $A^2$ & $A^3$ & $A^4$ & $A^5$\\\midrule
$I$ & $I$ & $A$ & $A^2$ & $A^3$ & $A^4$ & $A^5$\\
$A$ & $A$ & $A^2$ & $A^3$ & $A^4$ & $A^5$ & $I$\\
$A^2$ & $A^2$ & $A^3$ & $A^4$ & $A^5$ & $I$ & $A$\\
$A^3$ & $A^3$ & $A^4$ & $A^5$ & $I$ & $A$ & $A^2$\\
$A^4$ & $A^4$ & $A^5$ & $I$ & $A$ & $A^2$ & $A^3$\\
$A^5$ & $A^5$ & $I$ & $A$ & $A^2$ & $A^3$ & $A^4$\\\midrule
\end{tabular}}$

\smallskip
(c) Da. 


\smallskip
(d) $G=\langle A\rangle$, $G=\langle A^5\rangle$.




\begin{zad}
Naj bo $G=\Z_2\times\Z_3$ in $H=\Z_6$. Ali je $G\cong H$? Odgovor utemeljite!
\end{zad}


Re.

\smallskip
$G=\{(0,0),(0,1),(0,2),(1,0),(1,1),(1,2)\}$

\smallskip
${\it\begin{tabular}{c|cccccccc|}
\, & $(0,0)$ & $(0,1)$ & $(0,2)$ & $(1,0)$ & $(1,1)$ & $(1,2)$\\\midrule
$(0,0)$ & $(0,0)$ & $(0,1)$ & $(0,2)$ & $(1,0)$ & $(1,1)$ & $(1,2)$\\
$(0,1)$ & $(0,1)$ & $(0,2)$ & $(0,0)$ & $(1,1)$ & $(1,2)$ & $(1,0)$\\
$(0,2)$ & $(0,2)$ & $(0,0)$ & $(0,1)$ & $(1,2)$ & $(1,0)$ & $(1,1)$\\
$(1,0)$ & $(1,0)$ & $(1,1)$ & $(1,2)$ & $(0,0)$ & $(0,1)$ & $(0,2)$\\
$(1,1)$ & $(1,1)$ & $(1,2)$ & $(1,0)$ & $(0,1)$ & $(0,2)$ & $(0,0)$\\
$(1,2)$ & $(1,2)$ & $(1,0)$ & $(1,1)$ & $(0,2)$ & $(0,0)$ & $(0,1)$\\\midrule
\end{tabular}}$

\smallskip
$H=\{0,1,2,3,4,5\}$

\smallskip
${\it\begin{tabular}{c|cccccccc|}
\, & 0 & 2 & 4 & 3 & 5 & 1\\\midrule
0 & 0 & 2 & 4 & 3 & 5 & 1\\
2 & 2 & 4 & 0 & 5 & 1 & 3\\
4 & 4 & 0 & 2 & 1 & 3 & 5\\
3 & 3 & 5 & 1 & 0 & 2 & 4\\
5 & 5 & 1 & 3 & 2 & 4 & 0\\
1 & 1 & 3 & 5 & 4 & 0 & 2\\\midrule
\end{tabular}}$

\smallskip
Da, $G\cong H$, $\varphi:G\longrightarrow H$ je izomorfizem, kje $\varphi(0,0)=0$, $\varphi(0,1)=2$, $\varphi(0,2)=4$, $\varphi(1,0)=3$, $\varphi(1,1)=5$ in $\varphi(1,2)=1$...


\begin{zad}
Naj bo $S_4$ simetri\v{c}na grupa reda $4!$.

\smallskip
\hspace{5mm}
(40\%)(a) Dolo\v{c}ite podgrupo grupe $S_4$, ki vsebuje $6$ elementov.

\smallskip
\hspace{5mm}
(10\%)(b) Koliko podgrup reda $6$ obstaja v grupi $S_4$?

\smallskip
\hspace{5mm}
(50\%)(c) \v{C}e je $H=\langle(12),(13)\rangle$, dolo\v{c}ite vse leve odseke podgrupe $H$ v grupi $S_4$.
\end{zad}



Re.

\smallskip
(a) $S_3\le S_4$, $S_3=\{(1),(12),(13),(23),(123),(132)\}$.

\smallskip
(b) Obstajajo 4 grupe reda 6.

\smallskip
(c) $H=S_3$. Levi odseki so: $(1)H$, $(14)H$, $(24)H$, $(34)H$.






\begin{zad}
(50\%)(a) Naj bo $D_6$ diederska grupa reda $6$. Grupa $D_6$ deluje na mno\v{z}ici $X=\{A,B,C\}$ oglji\v{s}\v{c} enakostrani\v{c}nega trikotnika $\triangle ABC$. Dolo\v{c}ite orbito in stabilizator vsakega elementa iz mno\v{z}ice $X$ glede na delovanje grupe $D_6$.
\end{zad}
\vspace*{-6mm}
\hvFloat[%
onlyText=true,%
floatPos=htb,%
capWidth=1.0,% of \columnwidth
capPos=l,%
capVPos=c,%
objectPos=r]%
{}%
{\footnotesize\it\begin{tabular}{c|cccccccc|}
\, & $I$ & $A$ & $B$ & $AB$ & $BA$ & $ABA$\\\midrule
$I$ & $I$ & $A$ & $B$ & $AB$ & $BA$ & $ABA$\\
$A$ & $A$ & $I$ & $AB$ & $B$ & $ABA$ & $BA$\\
$B$ & $B$ & $BA$ & $I$ & $ABA$ & $A$ & $AB$\\
$AB$ & $AB$ & $ABA$ & $A$ & $BA$ & $I$ & $B$\\
$BA$ & $BA$ & $B$ & $ABA$ & $I$ & $AB$ & $A$\\
$ABA$ & $ABA$ & $AB$ & $BA$ & $A$ & $B$ & $I$\\\midrule
\end{tabular}}%
[e]{{
(50\%)(b) Mno\v{z}ica $G=\{I,A,B,AB,BA,ABA\}$ tvori grupo glede na operacijo mno\v{z}enja, in njena Cayley-eva tabela je podana na desni strani.\\
{\textcolor{white}{00}}
Dolo\v{c}ite vse Sylowe $2$-podgrupe grupe $G$ ter vse Sylowe $3$-podgrupe grupe $G$.
}}
{fig:text}




Re.

\smallskip
(a) $G=D_3=\{id,\rho,\rho^2,\sigma,\sigma\rho,\sigma\rho^2\}$, $GA=\{A,B,C\}=X$, $G_A=\{id,\sigma\}$, $G_B=\{id,\sigma\rho\}$, $G_C=\{id,\sigma\rho^2\}$. 

\smallskip
(b) Sylowe $2$-podgrupe grupe $G$ so $\{I,A\}$, $\{I,B\}$, $\{I,ABA\}$. Sylowe $3$-podgrupe grupe $G$ so $\{I,AB,BA\}$.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}