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group-theory

Normal subgroups

taught

Taught in

Requires

Required by

Local knowledge map

Local knowledge map

A first look at the First Isomorphism Theorem Abelianization Classification of finite abelian groups Groups of order p² are abelian Normal Sylow subgroups Quotient groups of cyclic groups Quotient groups of permutation groups Recognizing an internal direct product Small-order classification as a worked recipe Solvable groups The First Isomorphism Theorem (groups) The Jordan-Hölder theorem The Schur–Zassenhaus theorem The Third Sylow Theorem (counting) The quotient construction as a recurring pattern Characteristic subgroups Constructing a composition series Group extensions (introduced) Internal direct products Normal subgroups: examples and counterexamples Normality tests (generators, index 2) Quotient groups Simple groups: a first look The commutator subgroup The normalizer of a subgroup Well-definedness of coset multiplication Normal subgroups A first look at semidirect products Collapsing cosets into elements Coset representatives and the coset partition External direct products Groups of prime order are cyclic Index of a subgroup Inner automorphisms Outer automorphisms (introduced) The center of a group The conjugation action and conjugacy classes When left and right cosets agree Conjugation Left and right cosets Equivalence relations The group axioms The subgroup test

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Solid arrows are required prerequisites; dashed are recommended or helpful. Lower rows are more foundational. Click any box to jump to that concept.