\documentclass[12pt,a4paper]{article}

%%\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}
\usepackage[bookmarks=false]{hyperref}




\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}{250mm}
\setlength{\textwidth}{180mm}
\begin{document}
\raggedright




\renewcommand{\thesection}{}
\section{Algebra III - Abstraktna algebra, 01.02.2016.}
















\begin{zad}
Dana je množica $G=\{2x+3y\,\mid\,x-3y=0,\,x,y\in\Z\}$. 

\smallskip
\hspace{5mm}
(a.) Pokaži, da je $(G,+)$ grupa, kjer je $+$ običajno seštevanje celih števil. 

\smallskip
\hspace{5mm}
(b.) Dana je podgrupa $H=\{2x+3y\,\mid\,x-3y=0,\,y=2k,\,\,x,k\in\Z\}$ grupe $G$. Napiši Cayley-evo tabelo za $G/H$.
\end{zad}



Re.

\smallskip
(a.) $(2x_1+3y_1)+(2x_2+3y_2)=2(x_1+x_2)+3(y_1+y_2)$, $(x_1+x_2)-3(y_1+y_2)=0$, $0=2\cdot0+3\cdot 0$, $(2x+3y)+(2(-x)+3(-y))=0$.

\smallskip
(b.) $G=\{...,-45,-36,-27,-18,-9,0,9,18,27,36,45,...\}$, $H=\{...,-36,-18,0,18,36,...\}$

\medskip
\hspace{25mm}
{\begin{tabular}{c|cccccccc|}
$+$&$H$&$9+H$\\\midrule
$H$&$H$&$9+H$\\
$9+H$&$9+H$&$H$\\\midrule
\end{tabular}}
\medskip

\hfill{$\square$}







\begin{zad}
(a.) Dani sta grupi $(\Z_9,+)$ in $(\Z_3,+)$. Poišči vse homomorfizme iz grupe $\Z_9$ v grupo $\Z_3$.

\smallskip
\hspace{5mm}
(b) Naj bo $G=\Z_3\times\Z_3$ in $H=\Z_9$. Ali je $G\cong H$? Odgovor utemelji!
\end{zad}


Re.

\smallskip
(a.) $\phi_1(x)=0$, $\forall x\in\Z_9$; $\qquad\phi_2(x)=2x\,\mod\,3$, $\forall x\in\Z_9$.

\smallskip
(b.) $G\not\cong H$, $\forall(a,b)\in\Z_3\times\Z_3$ $|(a,b)|\le 3$.


\hfill{$\square$}






\vspace*{2mm}
\hvFloat[%
onlyText=true,%
floatPos=htb,%
capWidth=1.0,% of \columnwidth
capPos=l,%
capVPos=c,%
objectPos=r]%
{}%
{\it
\begin{tabular}{c|cccccccc|}
$\circ$&$1$&$a$&$b$&$c$&$d$&$e$&$f$&$g$\\\midrule
$1$&$1$&$a$&$b$&$c$&$d$&$e$&$f$&$g$\\
$a$&$a$&$e$&$c$&$g$&$b$&$f$&$1$&$d$\\
$b$&$b$&$c$&$f$&$1$&$e$&$g$&$d$&$a$\\
$c$&$c$&$g$&$1$&$a$&$f$&$d$&$b$&$e$\\
$d$&$d$&$b$&$e$&$f$&$a$&$c$&$g$&$1$\\
$e$&$e$&$f$&$g$&$d$&$c$&$1$&$a$&$b$\\
$f$&$f$&$1$&$d$&$b$&$g$&$a$&$e$&$c$\\
$g$&$g$&$d$&$a$&$e$&$1$&$b$&$c$&$f$\\\midrule
\end{tabular}
}%
[e]{{\begin{zad}
Množica $G=\{1,a,b,c,d,e,f,g\}$ tvori grupo glede na binarno operacijo $\circ$. Njena Cayley-eva tabela je dana na desni strani.\\
\smallskip
\hspace{5mm}
(a) Določi vse ciklične podgrupe grupe $G$. Za vsako ciklično podgrupo napiši vse njene generatorje.\\
\smallskip
\hspace{5mm}
(b) Za vsak $x\in G$ izračunaj $|x|$. Odgovor obrazloži!\\
\smallskip
\hspace{5mm}
(c.) Izračunaj center grupe $G$.\\
\smallskip
\hspace{5mm}
(d.) Izračunaj normalizator ter centralizator elementov $a$ in $g$.
\end{zad}}}
{fig:text}



Re.

\smallskip
(a.) $\langle1\rangle=\{1\}$, $\langle a\rangle=\{1,a,e,f\}$, $\langle e\rangle=\{1,e\}$, $\langle f\rangle=\{1,a,e,f\}$, $\langle b\rangle=\langle c\rangle=\langle d\rangle=\langle g\rangle=G$.

\smallskip
(b.) $|1|=1$, $|e|=2$, $|a|=|f|=4$, $|b|=|c|=|d|=|g|=8$.


\smallskip
(c.) $Z(G)=G$.


\smallskip
(d.) $N(a)=N(g)=G$.


\hfill{$\square$}








\begin{zad}
(a.) Naj bo $G$ podgrupa grupe $S_8$, generirana z elementoma $(123)(45)$ in $(78)$. Potem $G$ kot grupa permutacij deluje na množici $X=\{1,2,...,8\}$. Poišči orbito in stabilizator vseh elementov množice $X$.

\smallskip
\hspace{5mm}
(b) Naj bo $G$ poljubna grupa, ki deluje na neki množici $X$. Predpostavimo, da je pri tem delovanju 6 orbit. Naj bo $H\le G$ in naj bo $[G:H]=3$.
Koliko orbit ima lahko delovanje podgrupe $H$ na množici $X$?
\end{zad}

Re.

\smallskip
(a.) $G1=G2=G3=\{1,2,3\}$, $G4=G5=\{4,5\}$, $G6=\{6\}$, $G7=G8=\{7,8\}$; $G_1=G_2=G_3=G_4=G_5=\langle(78)\rangle$, $G_6=G$, $G_7=G_8=\langle(123)(45)\rangle$.


\smallskip
(b.) $|H|=\frac{1}{3}|Gx||G_x|$, $|H|\le|Hx||G_x|$. Delovanje podgrupe $H$ na množici $X$ ima lahko od 6 do 18 orbit.



\hfill{$\square$}






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