1. Groups and subgroups
- group
- abelian group
- examples of groups
- Z	group of integers (addition)
- K\_4	Klein fourgroup
- Dih(n)	Dihedral group of order n
- GL(n,R)	General linear group of R^n Euclidean space
- properties of groups
2. Subgroups
- subgroup
- examples of subgroups
- nZ	group of multiples of n (addition)
- O(n,R)	Orthogonal group of R^n Euclidean space
- properties of subgroups
- left coset
- right coset
- properties of cosets
- Lagrange's theorem

2. Group homomorphisms
- group homomorphism
- group monomorphism
- group epimorphism
- group isomorphism
- group endomorphism
- group automorphism
- properties of group homomorphisms
- properties of group isomorphisms

3. Quotient groups
- normal subgroup
- (canonical) quotient map (why we need normal subgroups)
- quotient group
- examples of quotient groups
- Z/nZ	group of integers modulo n (addition)
- T	Circle group
- homomorphism theorem
- first isomorphism theorem (group)
- second isomorphism theorem (group)

4. Cyclic groups
- cyclic group
- Cyc(n)	Cyclic group of order n
- generator element
- order
- cyclic subgroup
- properties of cyclic groups
- subgroups of cyclic groups are cyclic
- d divides the order of cyclic group implies unique subgroup of order d
- sum of Euler totients of divisors of N equals N 
- Cyclic group isomorphism to Z\nZ
- Cauchy's theorem

5. Product groups
- direct product group
- semidirect product group
- when is a direct-semidirect map an isomorphism? 
- chinese remainder theorem
- Sylow's theorem

6. Symmetric groups
- permutation function
- symmetric group 
- symmetric group 
- Sym(n)	Symmetric group of order n
- matrix notation
- cycle notation
- k-cycle
- transpostion (2-cycle)
- simple transpostion
- disjoint cycles
- commutativity of disjoint cycles
- all bijections have unique disjoint cycle product representation
- cycle type
- conjugate cycle lemma
- cycle type-conjugation theorem
- inversion set
- properites of inversion set cardinality
- permutation signature
- signature homomorphism
- alternating group
- Alt(n)	Alternating group
- simple group

- permutation group
- Cayley's theorem

7. Group actions
- group action







8. Group-like structures
- monoid
- magma

9. Lie theory




- Burnside's lemma