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

%%\usepackage[cp1250]{inputenc} 
\usepackage[cp1250]{inputenc} 
\usepackage[left=16mm,right=17mm,top=20mm,bottom=20mm]{geometry}
\usepackage{makeidx, graphicx, amsmath, amssymb, amsthm, ae}
\usepackage[croatian, english]{babel}
\usepackage{esint, hvfloat}
%%\usepackage{systeme,mathtools}
\usepackage{booktabs, multicol, color, url}




\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\id{{\mbox{\rm id}}}
\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, 11.01.2017.}}










\bigskip
\begin{zad}
Določi število elementov reda 12 v grupi $\Z_3\times\Z_4\times U(9)$.
\end{zad}

\medskip
Re.

\smallskip
$\Z_3=\{0,1,2\}$, $|0|=1$, $|1|=3$, $|2|=3$;

$\Z_4=\{0,1,2,3\}$, $|0|=1$, $|1|=4$, $|2|=2$, $|3|=4$;

$U(9)=\{1,2,4,5,7,8\}$, $|1|=1$, $|2|=6$, $|4|=3$, $|5|=6$, $|7|=3$, $|8|=2$;

$\forall (g_1,g_2,g_3)\in\Z_3\times\Z_4\times U(9)$ imamo $12=|(g_1,g_2,g_3)|={\rm lcm}(|g_1|,|g_2|,|g_3|)$ če in samo če $(|g_1|,|g_2|,|g_3|)\in\{(1,4,3),(1,4,6),(3,4,1),(3,4,2),(3,4,3),(3,4,6)\}$...

$32$ elementa



\bigskip
\begin{zad}
Uporabi prvi izrek o izomorfizmu, in pokaži, da je $\Z\times\Z/\langle(2,7)\rangle\cong\Z$.
\end{zad}


\medskip
Re.

\smallskip
Če je $\phi:\Z\times\Z\rightarrow\Z$ definirana z $\phi(a,b)=7a-2b$ potem je $\phi$ homomorfizem, $\phi(\Z\times\Z)=\Z$ in $\ker\phi=\langle(2,7)\rangle$...



\bigskip
\begin{zad}
Dan je pravilni petkotnik katerega, ogljišča so označena s števili 1, 2, 3, 4 in 5. Naj bo $G$ grupa vseh simetrij pravilnega petokotnika 12345 in naj bo $H=G_1$ ($H$ je stabilizator ogljišča $1$ v grupi $G$).
\begin{itemize}
\item[{\rm (a)}] Določi element $\alpha\in G$ za katerega je $\alpha(1)=4$. Poišči orbito ogljišča 1 glede na grupo $G$.
\item[{\rm (b)}] Določi element $\beta\in H$ za katerega je $\beta(2)=5$. Poišči orbito ogljišča 2 glede na grupo $H$. 
\item[{\rm (c)}] Ispiši vse elemente stabilizatora $H_2$. 
\item[{\rm (d)}] Uporabi orbita-stabilizator izrek in dokaži, da je $|G|=10$. 
\end{itemize}
\end{zad}


\medskip
Re.

\smallskip
$G=D_5$, $H=\{(1),(25)(34)\}$, $\alpha=(14)(23)$, $G1=\{1,2,3,4,5\}$, $\beta=(25)(34)$, $H2=\{2,5\}$, $H_2=\{(1)\}$...


\bigskip
\begin{zad}
Poišči mogoče število Sylowih 3-podgrup in Sylowih 7-podgrup v grupi reda 3087.
\end{zad}


\medskip
Re.

\smallskip
$|G|=3087=3^2\cdot 7^3$;

$1+3m$ deli $|G|$ za $m\in\{0,2,16,114\}$;...

$1+7m$ deli $|G|$ za $m=0$;...



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




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