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\date{Due: October 10, 2025}

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\noindent \textbf{Homework Assignment 2\hfill Math 369 - Introduction to Abstract Algebra}
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\noindent\textit{``In mathematics you don't understand things. You just get used to them.''}   \hfill --- John von Neumann 
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\noindent \textbf{Instructions:} Show all work clearly. Justify your answers with complete explanations. Late assignments will not be accepted without prior arrangement.

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\noindent \textbf{Problem 1.} Let $G = \{e, a, b, c, d, f\}$ be a group with the following Cayley table:

\[
\begin{array}{c|cccccc}
\cdot & e & a & b & c & d & f \\
\hline
e & e & a & b & c & d & f \\
a & a & b & e & f & c & d \\
b & b & e & a & d & f & c \\
c & c & d & f & e & a & b \\
d & d & f & c & b & e & a \\
f & f & c & d & a & b & e \\
\end{array}
\]

\begin{enumerate}
\item[(a)] Find the order of each element in $G$.
\item[(b)] Find all subgroups of $G$ and determine which ones are cyclic.
\item[(c)] Show that $G \cong S_3$ by finding an explicit isomorphism $\phi: G \to S_3$.
\item[(d)] Use your isomorphism to identify which elements of $G$ correspond to transpositions in $S_3$.
\end{enumerate}

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\noindent \textbf{Problem 2.} Consider the group $(\mathbb{Z}_{18}, +)$.

\begin{enumerate}
\item[(a)] Find all generators of $\mathbb{Z}_{18}$. Explain your reasoning.
\item[(b)] For each divisor $d$ of 18, find the unique subgroup of order $d$ in $\mathbb{Z}_{18}$.
\item[(c)] Draw the subgroup lattice for $\mathbb{Z}_{18}$, showing all subgroups and their inclusion relationships.
\item[(d)] Find all elements of order 6 in $\mathbb{Z}_{18}$.
\end{enumerate}

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\noindent \textbf{Problem 3.} Prove that every group of order 4 is isomorphic to $\mathbb{Z}_4$ or $V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$, where $V_4 = \{e, a, b, c\}$ with $a^2 = b^2 = c^2 = e$ and $ab = c$, $ac = b$, $bc = a$.

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\noindent \textbf{Problem 4.} Suppose $G$ is a finite group of odd order. Show that $f: G \to G$, $f(x) = x^2$ is an isomorphism if and only if $G$ is abelian.

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\noindent \textbf{Problem 5.} Find an isomorphism from the group $(\mathbb{R}\setminus\{0\}, \cdot)$ where $\cdot$ is multiplication to the group $(\mathbb{R}\setminus\{0\}, *)$ where $a * b = \frac{ab}{\pi}$.

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\noindent \textbf{Problem 6.} For each statement below, either prove it is true or provide a counterexample.

\begin{enumerate}
\item[(a)] If $G$ is an infinite abelian group, then $G$ is cyclic.
\item[(b)] If $\phi: G \to H$ is an isomorphism and $G$ is not cyclic, then $H$ is not cyclic.
\item[(c)] If $G$ is an abelian group of order 8, then $G$ is cyclic.
\item[(d)] If $G$ and $H$ are nonabelian groups with $|G| = |H|$, then $G \cong H$.
\end{enumerate}

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