Learn Year 2

Group theory

195 ideas in teaching order. Each idea comes after the ideas that it builds on. 194 of them have a page so far; the others are on the list in grey.

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Binary operation definition
Associativity definition
Commutativity definition
Identity element definition
Inverse element definition
Cayley table definition
Semigroup definition
Monoid definition
Group of units of a monoid definition
Group definition
Abelian group definition
Uniqueness of inverses statement
Cancellation laws statement
Laws of exponents in a group statement
Trivial group example
Additive group of integers example
Circle group object
Group of nth roots of unity object
Order of a group definition
Order of an element definition
g to the n equals 1 iff the order divides n statement
Order of an inverse statement
Order of a power statement
Subgroup definition
Subgroup criterion statement
Intersection of subgroups is a subgroup statement
Subgroup generated by a subset definition
Generators of a group concept
Finitely generated group definition
Lattice of subgroups concept
Cyclic subgroup generated by an element definition
Cyclic group definition
Cyclic group of order n example
Infinite cyclic group example
Cyclic groups are abelian statement
Order of a cyclic subgroup equals the order of its generator statement
Cyclic groups of the same order are isomorphic statement
Subgroups of cyclic groups are cyclic statement
Subgroups of the integers statement
Subgroups of a finite cyclic group correspond to divisors of its order statement
Number of elements of each order in a cyclic group statement
Product of cyclic groups is cyclic iff the orders are coprime statement
Left coset definition
Right coset definition
Cosets have the same cardinality as the subgroup statement
Index of a subgroup definition
Lagrange's theorem statement
Order of a group is the index times the order of the subgroup statement
Order of an element divides the order of the group statement
g to the order of the group is the identity statement
Groups of prime order are cyclic statement
Tower law for indices statement
Converse of Lagrange's theorem fails for the alternating group A4 counterexample
Product of two subgroups definition
Product formula for the order of HK statement
Conjugate elements definition
Conjugate subgroup definition
Normal subgroup definition
Normality via left and right cosets statement
Subgroups of abelian groups are normal statement
Subgroups of index two are normal statement
A subgroup whose index is the smallest prime divisor of the order is normal statement
Product of a subgroup and a normal subgroup is a subgroup statement
Normal core definition
Quotient group definition
Integers modulo n as a quotient group example
Group homomorphism definition
Homomorphisms preserve identity and inverses statement
Kernel of a homomorphism definition
Image of a homomorphism definition
The kernel is a normal subgroup statement
A homomorphism is injective iff its kernel is trivial statement
Group isomorphism definition
Determinant as a group homomorphism example
Universal property of the quotient group statement
First isomorphism theorem statement
Second isomorphism theorem statement
Third isomorphism theorem statement
Correspondence theorem statement
Automorphism group definition
Inner automorphism definition
Inner automorphism group is isomorphic to G modulo its centre statement
Outer automorphism group definition
Automorphism group of a cyclic group statement
Characteristic subgroup definition
Characteristic subgroups are normal statement
Centre of a group definition
Centraliser definition
Normaliser definition
If G modulo its centre is cyclic then G is abelian statement
Commutator of two elements definition
Commutator subgroup definition
A quotient is abelian iff the subgroup contains the commutator subgroup statement
Abelianisation definition
Universal property of the abelianisation statement
Symmetric group definition
Order of the symmetric group statement
Cycle notation concept
Transposition definition
Disjoint cycles commute statement
Decomposition of a permutation into disjoint cycles statement
Cycle type definition
Order of a permutation is the lcm of its cycle lengths statement
Conjugacy classes of the symmetric group correspond to cycle types statement
Decomposition of a permutation into transpositions statement
The symmetric group is generated by transpositions statement
Sign of a permutation definition
Sign is a homomorphism statement
Sign of a cycle statement
Even and odd permutations definition
Alternating group definition
Order of the alternating group statement
The alternating group is generated by three-cycles statement
Simplicity of the alternating group statement
The alternating group A5 is simple example
Symmetry group concept
Dihedral group definition
Order of the dihedral group statement
The symmetric group S3 example
The dihedral group of order 8 example
Quaternion group object
Klein four-group object
Order of the general linear group over a finite field statement
Direct product of groups definition
Internal direct product definition
Semidirect product definition
Group action definition
orbit definition
Orbits partition the set statement
Stabiliser definition
Fixed points of an action definition
Transitive action definition
Faithful action definition
Action by left multiplication example
Action by conjugation example
Action on left cosets example
Orbit-stabiliser theorem statement
Size of an orbit times the size of the stabiliser statement
Stabilisers of points in the same orbit are conjugate statement
Cayley's theorem statement
Burnside's lemma statement
Conjugacy class definition
Size of a conjugacy class is the index of the centraliser statement
Class equation statement
P-group definition
Fixed points of a p-group action statement
A nontrivial finite p-group has nontrivial centre statement
Groups of order p squared are abelian statement
Cauchy's theorem statement
Sylow p-subgroup definition
First Sylow theorem statement
Second Sylow theorem statement
Third Sylow theorem statement
The number of Sylow subgroups divides the index statement
The number of Sylow subgroups is the index of the normaliser statement
Frattini argument statement
Groups of order pq example
Simple group definition
Abelian simple groups have prime order statement
Subnormal series definition
Composition series definition
Composition factors definition
Jordan–Hölder theorem statement
Classification of finite simple groups statement
Solvable group definition
Derived series definition
Abelian groups are solvable statement
Subgroups of solvable groups are solvable statement
Quotients of solvable groups are solvable statement
Extensions of solvable groups are solvable statement
The symmetric group on five or more letters is not solvable statement
Nilpotent group definition
Lower central series definition
Upper central series definition
Finite p-groups are nilpotent statement
Nilpotent groups are solvable statement
A finite group is nilpotent iff it is the product of its Sylow subgroups statement
Burnside's p^a q^b theorem statement
Reduced word definition
Free group definition
Universal property of the free group statement
Free abelian group definition
Presentation of a group definition
Von Dyck's theorem statement
Cayley graph definition
Exponent of a group definition
A finite abelian group is cyclic iff its exponent equals its order statement
Torsion subgroup definition
Classification of finite abelian groups statement
Structure theorem for finitely generated abelian groups statement
Invariant factors and elementary divisors concept
Group of units of Z/nZ object
Order of the group of units of Z/nZ is Euler's totient statement
Units of Z/nZ are the residues coprime to n statement
The group of units of Z/pZ is cyclic statement