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\section*{Math 3140: Homework 8}


\begin{enumerate}
\item[A.]  \begin{enumerate}
\item[13.5]  Let $G$ be a group of order $4n+2$.  Show that $G$ contains a group of order $2n+1$.

Hint: Use Cauchy's Theorem, Cayley's Theorem, and Exercise 6.6.
\item[14.2] Find the conjugacy classes for $D_{n}$ for all $n$ (be careful to distinguish between different cases).
\item[14.3]  Suppose $\varphi:G\rightarrow H$ is an isomorphism of groups, and suppose $C$ is a conjugacy class of $G$.  Show that $\varphi(C)$ is a conjugacy class of $H$.
\item[14.5]  Prove that the $3$-cycles of $A_5$ form a single conjugacy class.  Find two $5$-cycles which are \emph{not} conjugate in $A_5$ (though they are conjugate in $S_5$).

\item[14.10] Find the center of $D_n$ for all $n$.

\end{enumerate}
\item[B.]  
\begin{enumerate}
\item Suppose $R$ and $S$ are rings.  Give a careful definition of what you think a \emph{ring homomorphism} should be.
\item Let $\varphi:R\rightarrow S$ be a ring homomorphism.  Define the \emph{kernel} of $\varphi$ to be the set
$$\mathrm{ker}(\varphi)=\{r\in R\mid \varphi(r)=0\}.$$
Show that for all $r\in R$ and $k\in \ker(\varphi)$, 
$$rk,kr\in \ker(\varphi).$$
\item An \emph{ideal} of a ring $R$ is a subring $I$ such that 
$$ri,ir\in I\qquad \text{for all $i\in I$, $r\in R$.}$$ 
Find a multiplicative function and an additive function that makes 
$$R+I=\{r+I\mid r\in R\}$$
a ring.
\item Show that $I\subseteq R$ is an ideal if and only if there exists a ring homomorphism $\varphi:R\rightarrow S$ for some ring $S$ such that $I=\ker(\varphi)$.
\end{enumerate}




\end{enumerate}

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