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\begin{document}




\raggedright
\renewcommand{\thesection}{}
\section{\centerline{Algebra III - Abstraktna algebra, 21.06.2016.}}
















\begin{zad}
Množica $G=\Z_4\times\Z_2$ tvori grupo glede na operacijo seštevanja.

\smallskip
\hspace{5mm}
(a) Napiši Cayley-evo tabelo grupe $G$.

\smallskip
\hspace{5mm}
(b) Za vse podgrupe reda $4$ v grupi $G$ napiši vse njihove generatorje.

\smallskip
\hspace{5mm}
(c) Poišči vse leve odseke podgrupe $H=\langle(1,1)\rangle$ v grupi $G$.
\end{zad}




Re.

(a)
$$
{\footnotesize\it\begin{tabular}{c|cccccccc|}
$+$&$(0,0)$&$(0,1)$&$(1,0)$&$(1,1)$&$(2,0)$&$(2,1)$&$(3,0)$&$(3,1)$\\\midrule
$(0,0)$&$(0,0)$&$(0,1)$&$(1,0)$&$(1,1)$&$(2,0)$&$(2,1)$&$(3,0)$&$(3,1)$\\
$(0,1)$&$(0,1)$&$(0,0)$&$(1,1)$&$(1,0)$&$(2,1)$&$(2,0)$&$(3,1)$&$(3,0)$\\
$(1,0)$&$(1,0)$&$(1,1)$&$(2,0)$&$(2,1)$&$(3,0)$&$(3,1)$&$(0,0)$&$(0,1)$\\
$(1,1)$&$(1,1)$&$(1,0)$&$(2,1)$&$(2,0)$&$(3,1)$&$(3,0)$&$(0,1)$&$(0,0)$\\
$(2,0)$&$(2,0)$&$(2,1)$&$(3,0)$&$(3,1)$&$(0,0)$&$(0,1)$&$(1,0)$&$(1,1)$\\
$(2,1)$&$(2,1)$&$(2,0)$&$(3,1)$&$(3,0)$&$(0,1)$&$(0,0)$&$(1,1)$&$(1,0)$\\
$(3,0)$&$(3,0)$&$(3,1)$&$(0,0)$&$(0,1)$&$(1,0)$&$(1,1)$&$(2,0)$&$(2,1)$\\
$(3,1)$&$(3,1)$&$(3,0)$&$(0,1)$&$(0,0)$&$(1,1)$&$(1,0)$&$(2,1)$&$(2,0)$\\\midrule
\end{tabular}}
$$

\smallskip
(b) $\langle (1,0)\rangle=\langle (3,0)\rangle=\{(0,0),(1,0),(2,0),(3,0)\}$,
$\langle (1,1)\rangle=\langle (3,1)\rangle=\{(0,0),(1,1),(2,0),(3,1)\}$,
$\langle (0,1),(2,0)\rangle=\langle (0,1),(2,1)\rangle=\{(0,0),(0,1),(2,0),(2,1)\}$.

\bigskip
(c) $H=\{(0,0),(1,1),(2,0),(3,1)\}$, $(0,1)+H=\{(0,1),(1,0),(2,1),(3,0)\}$.

\hfill{$\square$}








\begin{zad}
Naj bosta $\alpha=\left(\begin{matrix} 1&2&3&4&5&6\\2&1&3&5&4&6\end{matrix}\right)$ in $\beta=\left(\begin{matrix} 1&2&3&4&5&6\\6&1&2&4&3&5\end{matrix}\right)$. 

\hspace{5mm}
(a) Napiši $\alpha$, $\beta$ in $\alpha\beta$ kot produkt 2-ciklov (kot produkt transpozicij).

\hspace{5mm}
(b) Določi red permutacij $\alpha$ in $\beta$.

\hspace{5mm}
(c) Določi $\alpha^{-3}$.
\end{zad}


Re.

(a) $\alpha=(12)(45)$, $\beta=(12)(13)(15)(16)$, $\alpha\beta=(13)(15)(14)(16)$.

\bigskip
(b) $\alpha^2=id$, $|\alpha|=2$, $|\beta|=5$.

\bigskip
(c) $\alpha^{-3}=(12)(45)$.

\hfill{$\square$}




\begin{zad}
Naj bo $U(10) = \{ k \in \NN \mid 1 \le k \le 10, \; \gcd(k,10)=1\}$. Vemo, da je $U(10)$ grupa za množenje po modulu 10.

\smallskip
\hspace{5mm}
(a) Določi vse edinke grupe $U(10)$.\\
\smallskip
\hspace{5mm}
(b) Izračunaj center grupe $U(10)$.\\
\smallskip
\hspace{5mm}
(c) Določi vse homomorfizme iz grupe $U(10)$ v grupo $(\Z_8,+)$.
\end{zad}



Re.

(a) $\{1\}$, $\{1,9\}$ in $U(10)$.

\smallskip
(b) $Z(U(10))=U(10)$.

\smallskip
(c) Obstajajo $4$ homomorfizma. $\phi:U(10)\longrightarrow\Z_8$, $\phi(3^k)=ka$, kje je $a\in\{0,2,4,6\}$.

\hfill{$\square$}



\newpage
\begin{zad}
Za vsako od naslednjih delovanj grupe $G$ na množici $X$, opiši orbito in stabilizator danega elementa $x\in X$.

\smallskip
\hspace{5mm}
(a) $X=$ kvadrat, $G=\mbox{Sym}(X)$ in $x=$ ogljišče kvadrata.\\
\smallskip
\hspace{5mm}
(b) $X=\{1,2,3,4\}$, $G=A_4$ in $x=4$.\\
\smallskip
\hspace{5mm}
(c) $X=\R^2$, $G=\mbox{GL}_2(\R)$ in $x=(1,2)^\top$.
\end{zad}

Re.

\bigskip
(a) $X=\square ABCD$, $G=D_4$, $Gx=\{A,B,C,D\}$, $G_x=\{R_0, D'\}$.

\bigskip
(b) $Gx=\{1,2,3,4\}=X$, $G_x=\{id,(123),(132)\}$.

\bigskip
(c) $Gx=\R^2\backslash\{(0,0)^\top\}$, $G_x=\left\{\left(\begin{matrix} t&\frac{1}{2}(1-t)\\2-2s&s\end{matrix}\right):s,t\in\R,\, s+t-1\ne0\right\}$.

\hfill{$\square$}



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


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