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\begin{center}
\textbf{\LARGE{Oral Exam Syllabus}}

\textbf{\large{Laura Ciobanu}}
\end{center}
\section*{\underline{General/Combinatorial Group Theory}}

\subsection*{Free Groups and Their Subgroups}
\begin{description}
\item[-] Definition and elementary properties of free groups
\item[-] Nielsen-Schreier Theorem, Takahasi Theorem, Levi Theorem
\item[-] Properties of subgroups of finite index
\end{description}
{\small Ref: Magnus 1.4, 2.4; Lyndon 1.1, 1.3}


\subsection*{Automorphisms of Free Groups}
\begin{description}
\item[-] Presentation and properties of $Aut(F)$
\end{description}
{\small Ref: Lyndon 1.4; Magnus 3.5}


\subsection*{Nielsen Transformation}
\begin{description}
\item[-] Definition and applications
\item[-] Generalized word problem for (finitely generated) $H$ in $G$
\end{description}
{\small Ref: Magnus 3.1, 3.2; Lyndon 1.2}


\subsection*{Tietze Transformation}
\begin{description}
\item[-] Definition and applications
\end{description}
{\small Ref: Magnus 1.5}

\subsection*{Free Products, Free Products with Amalgamation, HNN Extensions}
\begin{description}
\item[-] Definitions and properties
\item[-] The normal forms for free products, free products with amalgamation and HNN~extensions
\item[-] The conjugacy theorems for free products, free products with amalgamation,
and HNN extensions
\item[-] The Higman, Neumann, and Neumann theorem
\end{description}
{\small Ref: Magnus 4.1, 4.2; Lyndon 1.11, 4.2}


\subsection*{Equations over Groups, Equations in Free Groups}

{\small Ref: Lyndon 1.6, 1.8}

\bigskip

\section*{\underline {Computational Group Theory}}

\subsection*{Rewriting Systems}
\begin{description}
\item[-] Orderings of free monoids, Canonical words, A test for confluence
\item[-] The Knuth-Bendix procedure
\item[-] Heuristics, Right Congruences
\end{description}
{\small Ref: Sims Ch. 2}

\subsection*{Subgroups of Free Products of Cyclic Groups}
\begin{description}
\item[-] Definition of languages and automata, Types of automata

\item[-] Niladic rewriting system, Subgroups and their languages, Important cosets,
Coset automata
\item[-] Basic coset enumeration
\item[-] The coincidence procedure, Standardization, Computation with subgroups, Standard coset tables  
\end{description}
{\small Ref: Sims Ch. 3, Ch. 4}

\subsection*{Coset Enumeration}
\begin{description}
\item[-] The general case
\item[-] The HLT strategy, The Felsch strategy, Standardizing strategies, Ten
  versions
\item[-] Low-index subgroups
\item[-] A comparison with the Knuth-Bendix procedure
\end{description}
{\small Ref: Sims Ch. 5}

\subsection*{The Reidemeister-Schreier Method}
\begin{description}
\item[-] Presentations of subgroups
\item[-] Examples of extended coset enumeration, An extended HLT enumeration
  procedure
\item[-] Symplifying presentations
\end{description}
{\small Ref: Sims Ch. 6, Magnus 2.3}

\subsection*{Abelian Groups}
\begin{description}
\item[-] Free abelian groups, Elementary matrices, Finitely generated abelian
  groups
\item[-] Modular techniques, The Kannan-Bachem algorithm
\item[-] Lattice reduction, The modified LLL algorithm, Comparison
\end{description}
{\small Ref: Sims Ch. 8}

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