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{\bf \Large 250A Homework 4}\\[7mm]
{\small Due Monday November 1}\\[4mm]
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\noindent
{\bf Question 1} 
Find all quotient groups for $D_8$. Draw the lattice
diagram and indicate which subgroups are normal. Also,
compute and compare all composition series of $D_8$.
The same for $S_4$.


\vspace{.4cm}\noindent
{\bf Question 2} 
Show that $H\unlhd G$ and $K\unlhd G$ $\Rightarrow$ $H\cap K\unlhd G$
and $HK\unlhd G$. Suppose the normal subgroups $K$ and $H$ are both maximal
in $G$, what does this imply for their product $KH$?

\vspace{.4cm}\noindent
{\bf Question 3} 
If $N\unlhd G$ and $G/N$ is an infinite cyclic group, prove that
$G$ has a normal subgroup of index $n$ $\forall n\in{\mathbb N}$.

\vspace{.4cm}\noindent
{\bf Question 4} 
Show that $Z_8$ has $Z_2$ and $Z_4$ as homomorphic images.
What can you say about the homomorphic images of cyclic groups?
What about abelian groups?


\vspace{.4cm}\noindent
{\bf Question 5}
Dcompose $A_4$ and $A_5$ into conjugacy classes. (Hint, what
happens to the cyclic form of a permutation under conjugation?)

\vspace{.4cm}\noindent
{\bf Question 6} Construct explicit examples of both 1st and 2nd
isomorphism theorems.

 
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