Math Problem Set

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\fancyhead[L]{\textbf{MATH 55a --- Honors Abstract Algebra}}
\fancyhead[R]{\textit{Problem Set 4}}
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  {\LARGE\bfseries MATH 55a --- Honors Abstract Algebra}\\[6pt]
  {\Large Problem Set 4}\\[8pt]
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\begin{tabular}{@{}ll}
  \textbf{Assigned:} & Tuesday, October 7, 2025 \\
  \textbf{Due:} & Tuesday, October 14, 2025, 11:59 PM \\
  \textbf{Instructor:} & Prof.\ David Kim \\
  \textbf{Course Assistants:} & Sarah Chen, Marcus Rivera
\end{tabular}

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\noindent\textbf{Instructions:}
\begin{itemize}[nosep,leftmargin=*]
  \item Write your solutions clearly and legibly. Justify all answers with complete proofs or explanations unless otherwise stated.
  \item You may consult the textbook (Artin, \textit{Algebra}, 2nd ed.) and lecture notes.
  \item Cite any theorems or lemmas you use from class or the text.
  \item \textbf{Collaboration policy:} You may discuss problems with classmates, but you must write up solutions independently and in your own words. List collaborators for each problem.
  \item Submit via Gradescope by the deadline. Late submissions receive a 10\% penalty per day.
\end{itemize}

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\noindent\textit{Collaborators:} \rule{0.6\textwidth}{0.4pt}

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\problem
Let $G$ be a group of order $p^2$ where $p$ is prime. Prove that $G$ is abelian.

\textit{Hint: Consider the center $Z(G)$ and use the class equation.}

\problem
Let $H$ and $K$ be subgroups of a finite group $G$.
\begin{enumerate}[label=(\alph*)]
  \item Prove that $|HK| = \frac{|H| \cdot |K|}{|H \cap K|}$.
  \item Give an example showing that $HK$ need not be a subgroup of $G$.
  \item Prove that if $[G : H] = 2$, then $H$ is normal in $G$.
\end{enumerate}

\problem
Let $\phi : G \to H$ be a group homomorphism.
\begin{enumerate}[label=(\alph*)]
  \item Prove that if $G$ is abelian, then $\phi(G)$ is abelian.
  \item Prove that if $G$ is cyclic, then $\phi(G)$ is cyclic.
  \item Show by example that the converse of (a) is false: $\phi(G)$ abelian does not imply $G$ abelian.
\end{enumerate}

\problem
Let $G$ be a finite group acting on a set $X$.
\begin{enumerate}[label=(\alph*)]
  \item State and prove the Orbit-Stabilizer Theorem: for any $x \in X$,
  \[
  |G| = |\text{Orb}(x)| \cdot |\text{Stab}(x)|.
  \]
  \item Use part (a) to count the number of rotational symmetries of a regular tetrahedron.
  \item Let $G = S_4$ act on $\{1, 2, 3, 4\}$ in the natural way. Find $\text{Orb}(1)$ and $\text{Stab}(1)$.
\end{enumerate}

\problem
\begin{enumerate}[label=(\alph*)]
  \item Classify all groups of order 6 up to isomorphism. \textit{(You may assume the Sylow theorems.)}
  \item Prove that every group of order 15 is cyclic.
  \item Determine the number of Sylow 3-subgroups and Sylow 5-subgroups of a group of order 15.
\end{enumerate}

\problem
Let $R$ be a commutative ring with identity.
\begin{enumerate}[label=(\alph*)]
  \item Prove that an ideal $I \subseteq R$ is maximal if and only if $R/I$ is a field.
  \item Show that $\langle x^2 + 1 \rangle$ is a maximal ideal in $\R[x]$. What is the quotient ring $\R[x]/\langle x^2 + 1 \rangle$ isomorphic to?
  \item Is $\langle x^2 + 1 \rangle$ a maximal ideal in $\Q[x]$? Prove or disprove.
\end{enumerate}

\problem
(\textbf{Challenge --- optional, for extra credit})

Let $G$ be a finite group and let $p$ be the smallest prime dividing $|G|$. Prove that any subgroup $H$ of $G$ with $[G : H] = p$ is normal in $G$.

\textit{Hint: Consider the action of $G$ on the left cosets $G/H$ by left multiplication, and analyze the resulting homomorphism $\phi : G \to S_p$.}

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  {\small\textit{End of Problem Set 4}}
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