Group Theory
Lagrange's Theorem
Subgroup order divides group order
Cayley's Theorem
Every group embeds into a symmetric group
Sylow Theorems
Prime-power subgroups always exist, are conjugate, and their count is constrained
First Isomorphism Theorem
The image of a homomorphism is isomorphic to the domain modulo the kernel
Cauchy's Theorem (Groups)
If a prime p divides |G| then G has an element of order p
Burnside's Lemma
The number of orbits equals the average size of fixed-point sets over the group.
Class Equation
The order of a finite group equals the size of its center plus the sum of sizes of nontrivial conjugacy classes.
FTAG (Finite Abelian)
Every finite abelian group decomposes as a direct sum of cyclic groups of prime power order.
Solvable Group
A group is solvable if its derived series reaches the trivial group in finitely many steps.
Nilpotent Group
A group is nilpotent if its lower central series reaches the trivial subgroup in finitely many steps.
Jordan-Hölder Theorem
Any two composition series of a group have the same length and isomorphic composition factors (in some order).
Semidirect Product
A group extension N ⋊ G where G acts on N by automorphisms, generalizing the direct product.
Second Isomorphism Theorem
For subgroups H and normal N of G, the quotient HN/N is isomorphic to H/(H ∩ N).
Third Isomorphism Theorem
A quotient of quotients collapses to a single quotient: (G/N)/(H/N) ≅ G/H
Correspondence Theorem
Subgroups of G/N are in bijection with subgroups of G containing N, preserving the lattice
Free Group
The free group on a set X is the universal group with generators X and no relations
Group Presentation
Every group can be described by generators and relations via a quotient of a free group
Simple Groups
A simple group has no proper nontrivial normal subgroups; the atoms of group composition
Conjugation Action
Every group acts on itself by conjugation; orbits are conjugacy classes
Alternating Group
The alternating group Aₙ consists of even permutations; it has order n!/2 for n ≥ 2
Dihedral Group
The dihedral group Dₙ describes the symmetries of a regular n-gon; it has order 2n