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\begin{document}




\raggedright


\renewcommand{\thesection}{}
\section{\centerline{Algebra III - Abstraktna algebra, 01.12.2017.}}








\vspace*{-1mm}
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{}%
{\footnotesize\begin{tabular}{c|cccccccc|}
* & A & B & C & D & P & Q & R \\\midrule
A & A & B & C & D & P & Q & R \\ 
B & B & ~ & ~ & ~ & ~ & R & ~ \\ 
C & C & ~ & ~ & ~ & R & ~ & ~ \\ 
D & D & ~ & ~ & R & ~ & ~ & ~ \\ 
P & P & ~ & R & ~ & ~ & ~ & ~ \\ 
Q & Q & R & ~ & ~ & ~ & D & P \\ 
R & R & ~ & ~ & ~ & ~ & P & Q \\\midrule
\end{tabular}}%
[e]{{
\begin{zad}
\begin{itemize}
\item[{\rm (a)}] Dana je končna grupa reda $n$. Pokaži, da se vsak element te grupe pojavi natanko  enkrat v vsaki vrstici in vsakem stolpcu Cayleyeve tabele te grupe.
\item[{\rm (b)}] Dana je množica ${\cal A}=\{A,B,C,D,P,Q,R\}$, in dana je binarna operacija $*$ definirana na množici ${\cal A}$, tako da je $({\cal A},*)$ grupa. Izpolni dano tabelo na desni strani tako, da bo ${\cal A}$ abelska\\ 
grupa.
\end{itemize}
\end{zad}
}}
{fig:text}



\medskip
Re. 

\smallskip
(a) Če je $ab=c$ in $ad=c$, potem $b=d$. Če je $ab=c$ in $gb=c$ potem $a=g$.


\smallskip
(b) Vsaka grupa praštevilskega reda je ciklična. Dana grupa je reda $7$.


{\footnotesize\begin{tabular}{c|cccccccc|}
* & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\midrule
0 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ 
1 & 1 & 2 & 3 & 4 & 5 & 6 & 0 \\ 
2 & 2 & 3 & 4 & 5 & 6 & 0 & 1 \\ 
3 & 3 & 4 & 5 & 6 & 0 & 1 & 2 \\ 
4 & 4 & 5 & 6 & 0 & 1 & 2 & 3 \\ 
5 & 5 & 6 & 0 & 1 & 2 & 3 & 4 \\ 
6 & 6 & 0 & 1 & 2 & 3 & 4 & 5 \\\midrule
\end{tabular}}\qquad$\cong$\qquad
{\footnotesize\begin{tabular}{c|cccccccc|}
* & A & B & C & D & P & Q & R \\\midrule
A & A & B & C & D & P & Q & R \\ 
B & B & C & D & P & Q & R & A \\ 
C & C & D & P & Q & R & A & B \\ 
D & D & P & Q & R & A & B & C \\ 
P & P & Q & R & A & B & C & D \\ 
Q & Q & R & A & B & C & D & P \\ 
R & R & A & B & C & D & P & Q \\\midrule
\end{tabular}}










\begin{zad}
Naj bo $G=(\Z_{126},+)$ dana grupa.
\begin{itemize}
\item[{\rm (a)}]Poišči vse možne rede elementov grupe $G$. Za vsakega od možnih redov podaj element grupe $G$, ki ima tak red. 
\item[{\rm (b)}] Za vsakega od možnih redov določi število elementov grupe $G$, ki imajo ta red. 
\item[{\rm (c)}] Določi število (cikličnih) podgrup grupe $G$, ki so reda $7$ (odgovor obrazloži brez uporabe znane trditve s predavanja ali vaj). 
\end{itemize}
\end{zad}


\medskip
Re. 

\smallskip
(a) Grupa G lahko ima elemente reda: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126. 
$$
|0|=1,\,\, |63|=2,\,\, |42|=3,\,\, |21|=6,\,\, |18|=7,\,\, |14|=9,\,\, |9|=14,\,\, |7|=18, 
$$
$$
|6|=21,\,\, |3|=42,\,\, |2|=63,\,\, |1|=126
$$

\smallskip
(b) $\phi(1)=|U(1)|=1$, $\phi(2)=|U(2)|=1$,
$\phi(3)=|U(3)|=2$, $\phi(6)=|U(6)|=2$,
$\phi(7)=|U(7)|=6$, $\phi(9)=|U(9)|=6$,
$\phi(14)=|U(14)|=6$, $\phi(18)=|U(18)|=6$,
$\phi(21)=|U(21)|=12$, $\phi(42)=|U(42)|=12$,
$\phi(63)=|U(63)|=36$, $\phi(126)=|U(126)|=36$.


\smallskip
(c) Če je $H=\{e,a,a^2,a^3,...,a^7\}$ ciklična grupa reda 7 potem, so vsi elementi $\{a,a^2,a^3,...,a^7\}$ reda 7. Kako je $\phi(7)=6$ to obstaja samo ena podgrupa grupe $G$ reda $7$.


\begin{zad}
Dana je simetrična grupa $S_5$ (grupa vseh permutacij množice $\{1,2,3,4,5\}$).
\begin{itemize}
\item[{\rm (a)}] Pokaži, da $S_5$ ni abelska. 
\item[{\rm (b)}] Določi število elementov reda $3$ grupe $S_5$. 
\item[{\rm (c)}] Določi število elementov reda $2$ grupe $S_5$ 
\item[{\rm (d)}] Določi redove vseh $5!=120$ elementov iz $S_5$. 
\end{itemize}
\end{zad}




\medskip
Re. 

\smallskip
(a) $(12), (13)\in S_5$, $(12)(13)=(132)\ne(123)=(13)(12)$.

\smallskip
(b) $\ds{\frac{5\cdot 4\cdot 3}{3}=20}$

\smallskip
(c) $\ds{\frac{5\cdot 4}{2}+\frac{5\cdot 4\cdot 3\cdot 2}{8}=10+15=25}$

\smallskip
(d) $1$, $2$, $3$, $4$, $5$ in $6$.








\begin{zad}
Naj bo $f:G\longrightarrow G'$ homomorfizem grup in $H\le G$. Pokaži, da je 
$$
f(H)\le G'.
$$
\end{zad}




\medskip
Re. 

\smallskip
$f(e)=e'$, $e'\in f(H)$, $a'\cdot b'=f(a)\cdot f(b)=f(ab)\in f(H)$, $a\cdot a^{-1}=e$, $f(a)\cdot f(a^{-1})=f(aa^{-1})=f(e)=e'$...





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




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