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\title{Math 120: Homework 4}
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\date{}
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\begin{itemize}[noitemsep]
  \item Section 3.1 \# 34, 36, 42
  \item Section 3.2 \# 8
  \item Section 3.3 \# 2 (Prove the bijection and statement (1) only), \#3
  \item Section 3.5 \# 1 ($\sigma,\tau$ only), 7
\end{itemize}

\bigbreak\noindent
{\textbf{Problem 3.1 \# 34.}} Let $D_{2 n} = \langle r, s|r^n = s^2 = 1, r s
= s r^{- 1} \rangle$.

(a) Prove that $\langle r^k \rangle$ is a normal subgroup of $D_{2 n}$.

(b) Prove that $D_{2 n} / \langle r^k \rangle \cong D_{2 k}$.

\bigbreak\noindent
{\textbf{Problem 3.1 \# 36.}} Prove that if $G / Z (G)$ is cyclic then $G$
is abelian.

\bigbreak\noindent
{\textbf{Problem 3.1 \# 42.}} Assume that $H$ and $K$ are normal subgroups
of $G$ with $H \cap K = 1$. Prove that $x y = y x$ for $x \in H$ and $y \in
K$.

\bigbreak\noindent
{\textbf{Problem 3.2 \# 8.}} Prove that if $H$ and $K$ are finite subgroups
of $G$ whose orders are relatively prime then $H \cap K = 1$.

\bigbreak\noindent
{\textbf{Problem 3.3 \#2.}} Prove the lattice isomorphism theorem. (Prove
the existence of the bijection and statement (1) only.)

\bigbreak\noindent
{\textbf{Problem 3.3 \#3.}} Prove that if $H$ is a normal subgroup of $G$
of prime index $p$ then for all $K \leqslant G$ either $K \leqslant H$ or $G =
H K$ and $[K : K \cap H] = p$.

\bigbreak\noindent
{\textbf{Problem 3.5 \#1.}} Let 
\[\begin{array}{l}
\sigma=(1, 13, 5, 10)(3, 15, 8)(4, 14, 11, 7, 12, 9)\\
\tau=(1, 14)(2, 9, 15, 13, 4)(3, 10)(5, 12, 7)(6)(8, 11)\end{array}\]
in $S_{15}$.
(See Exercises~1 and~2 of Section~1.3.) Write each of $\sigma$ and $\tau$ as a
product of transpositions. Determine wheof these is an even permutation and
which is odd.

\bigbreak\noindent
{\textbf{Problem 3.5 \#7.}} Prove that the group of rigid motions of the
tetrahedron is isomorphic to~$A_4$.

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