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\section*{Math 3140: Homework 12}





\begin{enumerate}
\item[A.]
\begin{enumerate}
\item[18.1] Use orbit counting methods to find the number of distinct ways to paint the edges of a cube with two colors.  
\item[18.7.]  How many different ways are there of coloring the vertices and edges of a regular hexagon using red, black or yellow for the edges and black or white for the vertices?
\end{enumerate}
\item[B.]
\begin{enumerate}
\item[20.1] \begin{enumerate}
\item[(a)] Show that if $|G|=126$, then $G$ has a nontrivial proper normal subgroup.
\item[(b)] Show that if $|G|=1000$, then $G$ is not simple.
\item[(c)] Suppose $|G|=p^km$ where $p$ is prime and $p$ does not divide $m$. Prove that if $p>m$, then $G$ is not simple.
\end{enumerate}
\item[(1)] Prove that if $G$ is abelian and simple, then $G\cong \ZZ_p$ for some prime number $p$.
\item[20.3.] \begin{enumerate}
\item[(a)] Prove that if all the Sylow subgroups are normal, then $G$ is isomorphic to the direct product of of its Sylow subgroups.
\item[(b)] If you know that $G$ is abelian, and $|G|=154000$, then what do you know about $G$?
\end{enumerate}
\item[20.7.]  Classify the groups of order $p^2q$ if $p$ is not congruent to $\pm 1$ modulo $q$ (and $p\neq q$ are prime).
\end{enumerate}

\end{enumerate}


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