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\begin{document}
\title{Abstract Algebra summary}
\maketitle
\begin{enumerate}
\item Recap on sets, functions and relations
\begin{enumerate}
\item Standard notation
\item Functions; restriction, associativity of composition
\item Images and pre-images; properties
\item Checking a function is well-defined
\item Equivalence relations and classes; partitioning
\item Introduction to groups
\end{enumerate}
\item Permutation groups
\begin{enumerate}
\item Permutations; grouphood of Sym(S)
\item Cycle decomposition (existence and uniqueness); orbits
\item Examples. Isomorphism of $D_3$ to $S_3$
\item Order of a permutation; order as LCM of cycle lengths
\item Parity, with well-definedness; elements of $A_7$
\item Conjugates of permutations. Conjugacy $\Leftrightarrow$ same type.
\end{enumerate}
\item Groups
\begin{enumerate}
\item Definitions and notation
\item Basic consequences of axioms:
\begin{itemize}
\item Uniqeness of identity and inverse
\item Generators, cyclic groups, order of elements
\item $g_r=e \:\Rightarrow\: o(g)|r$
\item Two facts about finite cyclic groups
\item Conjugate elements and conjugacy classes. Example: $D_6$
\end{itemize}
\item Isomorphisms
\begin{itemize}
\item Preservation of identity and inverse
\item Isomorphism of cyclic groups to $\mathbf{Z}$
\end{itemize}
\item Creating new groups. Intersections
\item Subgroup lattices and partial orderings
\end{enumerate}
\item Cyclic groups
\begin{enumerate}
\item Basic theory. Isomorphism to $\mathbf{Z}$, cyclicity of subgroups
\item Infinite cyclic groups; adding same. Prime multiplication groups
\item Finite cyclic groups. Properties. $\zahl_{pq} \cong \zahl_p \times 
\zahl_q$
\end{enumerate}
\item Lagrange's theorem
\begin{enumerate}
\item Statement. Consequence (prime order groups are cyclic)
\item Cosets
\item Properties of cosets: other defn., partitioning, bijection
\item groups of order $\le 7$. 4 and 6 in gory detail.
\end{enumerate}
\item Normal subgroups and quotient groups
\begin{enumerate}
\item Normal subgroups. Equivalent definitions. $H \lhd G$ if $G$ contains 2
cosets of $H$.
\item Quotient groups
\item Examples of same
\end{enumerate}
\item Homomorphisms
\begin{enumerate}
\item Basic facts. Kernel (is a normal sg, and converse) and image (sg)
\item Examples
\item Isomorphism theorem
\item Examples: $\mathrm{Aut}(S_3) \cong S_3$, homs from $S_4$ to $A_4$, $S_3$
\item Quotient groups via congruences
\item Epilogue
\end{enumerate}
\end{enumerate}
\end{document}