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\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\,}}}



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\begin{document}
\raggedright




\renewcommand{\thesection}{}
\section{Algebra III - Abstraktna algebra, 2. kolokvij, 21.01.2016.}
















\begin{zad}
Naj bo 
$
   U(20) = \{ i \in \NN \mid 1 \le i \le 20, \; \gcd(i,20)=1\}.
$
Vemo, da je $U(20)$ grupa za množenje po modulu 20. Naj bo $H=\langle 9 \rangle$
ciklična podgrupa grupe $U(20)$, generirana z $9$. 

\smallskip
\hspace{5mm}
(a.) Poišči vse desne odseke podgrupe $H$ v grupi $U(20)$.

\smallskip
\hspace{5mm}
(b.) Napiši Cayley-evo tabelo za $U(20)/H$.

\smallskip
\hspace{5mm}
(c.) Poišči vse podgrupe grupe $U(20)/H$.
\end{zad}



Re.

\smallskip
(a.) Vsi desni odseki podgrupe $H$ v grupi $U(20)$ so $\{1,9\}$, $\{3,7\}$, $\{11,19\}$ in $\{13,17\}$.

\smallskip
(b.) $U(20)/H=\{1H,3H,11H,13H\}$.

\medskip
\hspace{25mm}
{\begin{tabular}{c|cccccccc|}
$\cdot$&$1H$&$3H$&$11H$&$13H$\\\midrule
$1H$&$1H$&$3H$&$11H$&$13H$\\
$3H$&$3H$&$1H$&$13H$&$11H$\\
$11H$&$11H$&$13H$&$1H$&$3H$\\
$13H$&$13H$&$11H$&$3H$&$1H$\\\midrule
\end{tabular}}
\medskip



\smallskip
(c.) Vse podgrupe grupe $U(20)/H$ so $\{1H\}$, $\{1H,3H\}$, $\{1H,11H\}$, $\{1H,13H\}$ in $U(20)/H$.

\hfill{$\square$}




\medskip
\medskip\medskip
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{\footnotesize\it\begin{tabular}{c|cccccccc|}
$*$&$1$&$-1$&$i$&$-i$&$j$&$-j$&$k$&$-k$\\\midrule
$1$&$1$&$-1$&$i$&$-i$&$j$&$-j$&$k$&$-k$\\
$-1$&$-1$&$1$&$-i$&$i$&$-j$&$j$&$-k$&$k$\\
$i$&$i$&$-i$&$-1$&$1$&$k$&$-k$&$-j$&$j$\\
$-i$&$-i$&$i$&$1$&$-1$&$-k$&$k$&$j$&$-j$\\
$j$&$j$&$-j$&$-k$&$k$&$-1$&$1$&$i$&$-i$\\
$-j$&$-j$&$j$&$k$&$-k$&$1$&$-1$&$-i$&$i$\\
$k$&$k$&$-k$&$j$&$-j$&$-i$&$i$&$-1$&$1$\\
$-k$&$-k$&$k$&$-j$&$j$&$i$&$-i$&$1$&$-1$\\\midrule
\end{tabular}}%
[e]{{\begin{zad}
Množica $G=\{1,-1,i,-i,j,-j,k,-k\}$ tvori grupo glede na operacijo množenja, in njena Cayley-eva tabela je dana na desni strani.\\
\smallskip
\hspace{5mm}
(a.) Določi vse edinke grupe $G$.\\
\smallskip
\hspace{5mm}
(b.) Izračunaj center grupe $G$.\\
\smallskip
\hspace{5mm}
(c.) Izračunaj normalizator ter centralizator elementov $j$ in $-k$.
\end{zad}}}
{fig:text}



Re.

\smallskip
(a.) Vse edinke grupe $G$ so $\{1\}$, $\{1,-1\}$, $\{1,-1,i,-i\}$, $\{1,-1,j,-j\}$, $\{1,-1,k,-k\}$ in $\{1,-1,i,-i,j,-j,k,-k\}$.

\smallskip
(b.) $Z(G)=\{1,-1\}$.

\smallskip
(c.) $N(j)=\{1,-1,j,-j\}$, $N(-k)=\{1,-1,k,-k\}$.

\hfill{$\square$}




\medskip
\begin{zad}
Naj bo $G$ grupa realnih števil z operacijo seštevanja $(\R,+)$.  Za $r \in G$ ter $(x,y) \in \R^2$ definirajmo
$
  r*(x,y)=(x+ry, y).
$
Naj bo $T$ poljubna točka v ravnini.

\smallskip
\hspace{5mm}
(a.) Pokaži, da je preslikava $*: G \times \R^2 \to \R^2$ delovanje grupe $G$ na množici $\R^2$.

\smallskip
\hspace{5mm}
(b) Geometrijsko opiši orbito, ki vsebuje točko $T$.

\smallskip
\hspace{5mm}
(c) Poišči stabilizator $G_T$.

\smallskip
\hspace{5mm}
(d) Če je $H=\{(1,2),(2,3)\} \subseteq \R^2$, določi $5+G_H$.
\end{zad}


Re.

\smallskip
(a.) $0*(x,y)=(x,y)$, $(r+s)*(x,y)=r*(s*(x,y))$.

\smallskip
(b.) $G_T=\{(x+ry,y):r\in\R\}$, premica skozi točko T vzporedna s x-osjo.

\smallskip
(c.) $G_T=\{0\}$.

\smallskip
(d.) $5+G_H=\{5\}$.

\hfill{$\square$}




\medskip
\begin{zad}
(a) Poišči vse Sylowe 2-podgrupe grupe $D_{10}$ (diederska grupa vseh simetrij pravilnega $10$-kotnika glede na operacijo kompozicije);

\smallskip
\hspace{5mm}
(b) Pokaži, da grupa $G$ reda 992 ne more biti enostavna.
\end{zad}




Re.

\smallskip
(a.) $D_{10}=\{R_0, R_{36}, R_{36}^2, R_{36}^3, R_{36}^4, R_{36}^5, R_{36}^6, R_{36}^7, R_{36}^8, R_{36}^9, F, R_{36}F,$ $R_{36}^2F, R_{36}^3F, R_{36}^4F, R_{36}^5F, R_{36}^6F, R_{36}^7F, R_{36}^8F, R_{36}^9F,\}$, $|D_{10}|=20$. Vse Sylowe 2-podgrupe grupe $D_{10}$ so $\{R_0, R_{36}^5, F, R_{36}^5F\}$, $\{R_0, R_{36}^5, R_{36}F, R_{36}^6F\}$, $\{R_0, R_{36}^2F, F, R_{36}^7F\}$, $\{R_0, R_{36}^3F, F, R_{36}^8F\}$ in $\{R_0, R_{36}^4F, F, R_{36}^9F\}$.


\smallskip
(b.) $992=2^5\cdot 31$, $n_2\in\{1,31\}$, $n_{31}\in\{1,2^5\}$...

\hfill{$\square$}

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