\contentsline {part}{I\hspace {1em}Fundamentals}{1}{}%
\contentsline {chapter}{\numberline {1}Groups and Subgroups}{3}{}%
\contentsline {section}{\numberline {1.1}Groups}{3}{}%
\contentsline {subsection}{\numberline {1.1.1}Properties of Groups}{5}{}%
\contentsline {subsection}{\numberline {1.1.2}Abelian groups}{6}{}%
\contentsline {subsection}{\numberline {1.1.3}Cayley table and group presentations}{6}{}%
\contentsline {subsection}{\numberline {1.1.4}Examples of Groups}{7}{}%
\contentsline {section}{\numberline {1.2}Subgroups}{7}{}%
\contentsline {subsection}{\numberline {1.2.1}Examples of Subgroups}{9}{}%
\contentsline {subsection}{\numberline {1.2.2}Properties of Subgroups}{9}{}%
\contentsline {section}{\numberline {1.3}Cosets}{9}{}%
\contentsline {subsection}{\numberline {1.3.1}Properties of Cosets}{10}{}%
\contentsline {subsection}{\numberline {1.3.2}Examples of cosets}{11}{}%
\contentsline {subsection}{\numberline {1.3.3}Lagrange's theorem}{12}{}%
\contentsline {section}{\numberline {1.4}Conjugacy}{12}{}%
\contentsline {subsection}{\numberline {1.4.1}Commutator}{12}{}%
\contentsline {chapter}{\numberline {2}Group homomorphisms}{13}{}%
\contentsline {section}{\numberline {2.1}Group homomorphism}{13}{}%
\contentsline {subsection}{\numberline {2.1.1}Examples of Group homomorphism}{14}{}%
\contentsline {section}{\numberline {2.2}Group isomorphism}{14}{}%
\contentsline {subsection}{\numberline {2.2.1}Properties of Group isomorphism}{15}{}%
\contentsline {subsection}{\numberline {2.2.2}Kernel of a group homomorphism}{15}{}%
\contentsline {subsection}{\numberline {2.2.3}Examples of isomorphisms}{16}{}%
\contentsline {chapter}{\numberline {3}Quotient Groups}{17}{}%
\contentsline {section}{\numberline {3.1}Normal subgroups}{17}{}%
\contentsline {section}{\numberline {3.2}Quotient groups}{19}{}%
\contentsline {subsection}{\numberline {3.2.1}Examples of quotient groups}{20}{}%
\contentsline {subsection}{\numberline {3.2.2}First isomorphism theorem}{20}{}%
\contentsline {chapter}{\numberline {4}Cyclic Groups}{23}{}%
\contentsline {section}{\numberline {4.1}Cyclic Groups}{23}{}%
\contentsline {section}{\numberline {4.2}Isomorphisms of Cyclic Groups}{24}{}%
\contentsline {section}{\numberline {4.3}Properties of Cyclic Groups}{24}{}%
\contentsline {chapter}{\numberline {5}Product Groups}{27}{}%
\contentsline {section}{\numberline {5.1}Direct Product Groups}{27}{}%
\contentsline {section}{\numberline {5.2}Product of group subsets}{28}{}%
\contentsline {subsection}{\numberline {5.2.1}Product of subgroups}{28}{}%
\contentsline {subsection}{\numberline {5.2.2}Second isomorphism theorem}{29}{}%
\contentsline {subsection}{\numberline {5.2.3}Third isomorphism theorem}{29}{}%
\contentsline {subsection}{\numberline {5.2.4}Product of subgroups and direct product}{30}{}%
\contentsline {section}{\numberline {5.3}Inner semidirect product group}{31}{}%
\contentsline {section}{\numberline {5.4}Outer semidirect product group}{31}{}%
\contentsline {chapter}{\numberline {6}Symmetric Group}{33}{}%
\contentsline {section}{\numberline {6.1}Cayley's theorem}{35}{}%
\contentsline {section}{\numberline {6.2}Cycles}{35}{}%
\contentsline {subsection}{\numberline {6.2.1}Properties of Cycles}{36}{}%
\contentsline {section}{\numberline {6.3}Inversions}{37}{}%
\contentsline {section}{\numberline {6.4}Alternating group}{37}{}%
\contentsline {subsection}{\numberline {6.4.1}Permutation signature}{37}{}%
\contentsline {subsection}{\numberline {6.4.2}Alternating group}{38}{}%
\contentsline {section}{\numberline {6.5}Simple group}{39}{}%
\contentsline {chapter}{\numberline {7}Group Actions}{41}{}%
\contentsline {section}{\numberline {7.1}Group Actions}{41}{}%
\contentsline {section}{\numberline {7.2}Orbits}{43}{}%
\contentsline {section}{\numberline {7.3}Stabilizers}{44}{}%
\contentsline {section}{\numberline {7.4}Orbit-Stabilizer theorem}{46}{}%
\contentsline {subsection}{\numberline {7.4.1}Burnside's lemma}{47}{}%
\contentsline {section}{\numberline {7.5}Conjugacy}{47}{}%
\contentsline {section}{\numberline {7.6}Groups of order $p^r$}{49}{}%
\contentsline {section}{\numberline {7.7}Sylow theorems}{49}{}%
\contentsline {section}{\numberline {7.8}Wreath product}{50}{}%
\contentsline {chapter}{\numberline {8}Group-like Structures}{51}{}%
\contentsline {section}{\numberline {8.1}Monoids}{51}{}%
\contentsline {section}{\numberline {8.2}Magmas}{52}{}%
\contentsline {section}{\numberline {8.3}Loops}{52}{}%
\contentsline {part}{II\hspace {1em}Advanced}{53}{}%
\contentsline {chapter}{\numberline {9}Free groups}{55}{}%
\contentsline {section}{\numberline {9.1}Tietze transformations}{55}{}%
\contentsline {section}{\numberline {9.2}Lattices}{55}{}%
\contentsline {chapter}{\numberline {10}Torsion elements}{57}{}%