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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\A{{\mbox{$\mathcal{A}$}}}
\def\O{{\mbox{$\mathcal{O}$}}}

\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\lcm{{\mbox{\rm lcm}}}

\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\J{{\mbox{$\pmb{J}$}}}
\def\I{{\mbox{$\pmb{I}$}}}
\def\j{{\mbox{$\pmb{j}$}}}
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\def\algmult{{\mbox{\rm alg\,mult}}}
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\def\char{{\mbox{\rm char}}}
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\def\dx{{\mbox{\rm \,d$x$}}}
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\def\dy{{\mbox{\rm \,d$y$}}}
\def\dz{{\mbox{\rm \,d$z$}}}
\def\tg{{\mbox{\rm tg\,}}}
\def\arctg{{\mbox{\rm arc\,tg\,}}}
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\begin{document}




\raggedright


\renewcommand{\thesection}{}
\section{\centerline{Algebra III - Abstraktna algebra, 24.01.2017.}}










\bigskip

\begin{zad}
\begin{itemize}
\item[{\rm (a)}] Naj bo $X$ neprazna množica. Potenčno množico $\P(X)$ opremimo z operacijo $\backslash$ (razlika množic)
$$
\A \backslash \B:=\{x\,|\,x\in\A\,\mbox{ in }\,x\not\in\B\},\quad\forall\A,\B\in\P(X).
$$
Ali je množica $\P(X)$ zaprta glede na operacijo $\backslash$? Ali je operacija $\backslash$ asociativna na množici $\P(X)$? Odgovor utemelji! {\hfill\small(50\%)}
\item[{\rm (b)}] 
Naj bo $\pi=\left(\begin{matrix} 
1&2&3&4&5&6&7&8&9\\ 
7&3&6&2&8&4&9&5&1
\end{matrix}\right)$. Izračunaj $\pi^{-1}$, $\pi^{2017}$ in $\pi^{-2}\pi^4$. Napiši $\pi^{-2}\pi^4$ kot produkt 2-ciklov (torej kot produkt transpozicij).
{\hfill\small(50\%)}
\end{itemize}
\end{zad}


\medskip
Re. 

\smallskip
(a) je zaprta, ni asocijativna. Npr če je $\A:=\{a\}\subset\P(X)$ in $\B:=\emptyset\subset\P(X)$ potem $(\A\backslash\B)\backslash\A=\emptyset$, $\A\backslash(\B\backslash\A)=\A$...

\smallskip
(b) $\pi=(179)(2364)(58)$, $\pi^{-1}=(971)(4632)(58)$, $|\pi|=\lcm(3,4,2)=12$, $\pi^{12}=\id$, $\pi^{2017}=\pi$, $\pi^{-2}\pi^4=\pi^2=(17)(19)(26)(34)$.



\bigskip
\begin{zad}
Določi vse elemente reda $9$ v grupi $\Z_{108}$. Za vse podgrupe reda $9$ v grupi $\Z_{108}$ napiši vse njihove generatorje.
\end{zad}


\medskip
Re. 

\smallskip
Uporabi izrek: Za vsak pozitiven delitelj $k$ števila $n$, je množica $\langle n/k\rangle$ podgrupa grupe $\Z_n$ reda $k$. Poleg tega, te podgrupe so edine podgrupe grupe $\Z_n$;

ali izrek za število elementov reda $d$ v ciklični grupi: Če je $d$ pozitivno celo število ki deli $n$, potem je število elementov reda $d$ v ciklični grupi reda $n$ eneko $\phi(d)$;

ali fundamentalni izrek za ciklične grupe...

$|12|=|24|=|48|=|60|=|84|=|96|=9$.



\bigskip
\begin{zad}
\begin{itemize}
\item[{\rm (a)}] Dana je grupa $G=\{
1,8,12,14,18,21,27,31,34,38,44,47,51,53,57,64
\}$ za operacijo množenja po modulu $65$, in naj bo $H=\langle 12\rangle$ podgrupa grupe $G$. Napiši Cayley-evo tabelo za $G/H$.
\item[{\rm (b)}] Določi red elementa $8\langle16\rangle$ v grupi $U(105)/\langle16\rangle$.
\end{itemize}
\end{zad}


\medskip
Re.

\smallskip
(a)\qquad
\begin{tabular}{ c | c  c c c }			
  $\cdot$ & $H$ & $8H$ & $18H$ & $27H$ \\
    \hline
   $H$ & $H$ & $8H$ & $18H$ & $27H$\\
   $8H$ & $8H$ & $27H$ & $H$ & $18H$ \\
   $18H$ & $18H$ & $H$ & $27H$ & $8H$ \\
   $27H$ & $27H$ & $18H$ & $8H$ & $H$ \\
\end{tabular}

\bigskip
(b) $|8\langle 16\rangle|=4$.





\newpage
\begin{zad}
Naj bo $\O$ grupa vseh simetrij kocke (rotacija, zrcaljenje, drsno zrcaljenje,...). Grupa $\O$ deluje na množici $\{v_1,v_2,...,v_8\}$ ogljišč kocke. Določi stabilizator ogljišča $v_1$ v grupi $\O$. Uporabi orbita-stabilizator izrek in dokaži, da je $|\O|=48$.
\end{zad}


\medskip
Re.

\smallskip
\begin{center}
\includegraphics[scale=0.3]{02}
\end{center}
Če so $\{1,2,3,4,5,6,7,8\}$ ogljišč kocke potem
$$
\O_1=\{(1),(254)(368),(245)(386),(25)(38),(36)(45),(24)(68)\}.
$$
Če so $\{v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8\}$ ogljišč kocke potem

$|\O|=|\O v_1|\cdot|\O_{v_1}|=8|\O_{v_1}|$,

$|\O_{v_1}|=|\O_{v_1} v_2|\cdot|\O_{v_1v_2}|=3|\O_{v_1v_2}|$,

$|\O_{v_1v_2}|=|\O_{v_1v_2}v_4|\cdot|\O_{v_1v_2v_4}|=2|\O_{v_1v_2v_4}|$,...




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




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