This is a definition in Abstract Algebra that is used to help investigate similar patterns and properties across various mathematical objects.

Definition

A group is a set $G$ together with an operation $*$ that combines any two elements in the set to form a third element also in the set.

This operation, applied to any two elements, keeps us in the set $G$

The Four Axioms

These are fundamental truths of groups. It is the starting point for arguments and logical deductions.

Closure For all $a, b \in G$ , the result $a * b \in G$ always
Associativity For all $a, b \in G$ ,

(a * b) * c = a * (b * c)

Identity There exists an element $e \in G$ such that for all elements $a \in G$ ,

a * e = e * a = a

Inverse For every $a \in G$ , there exists an “inverse” element $a^{- 1} \in G$ such that

a * a^{- 1} = a^{- 1} * a = e

EXAMPLE (Integers under addition) This group is denoted as

(Z, +)

Closure: $3 + 5 = 8$ ✓
Associativity: $(1 + 2) + 3 = 1 + (2 + 3) = 6$ ✓
Identity: $0$ (since $a + 0 = a$ ) ✓
Inverse: $- a$ for each $a$ (since $a + (- a) = 0$ ) ✓

This means that Integers under addition is considered a group.

When you define a group, and that group follows the four axioms. Then that group you made follows over 150+ properties! This is because these properties are derived from logical deductions from the 4 axioms.

Free Access to Properties

There are a ton of properties that have been proven in mathematics just from logical deductions from The Four Axioms above.

Here’s a non-exhaustive list of them (LLM generated lol):

Foundational Properties (10)

Unique identity
Unique inverses
$(a^{- 1})^{- 1} = a$
$(ab)^{- 1} = b^{- 1} a^{- 1}$
Left cancellation law
Right cancellation law
$a x = b$ has unique solution $x = a^{- 1} b$
$y a = b$ has unique solution $y = b a^{- 1}$
$e^{- 1} = e$
$(e * a = a)$ and $(a * e = a)$ define same $e$

Exponent/Power Laws (8)

$a^{m} a^{n} = a^{m + n}$
$(a^{m})^{n} = a^{mn}$
$(a^{- 1})^{n} = (a^{n})^{- 1} = a^{- n}$
If $a^{n} = e$ , then $(a^{- 1})^{n} = e$
$a^{0} = e$ by definition
In abelian groups: $(ab)^{n} = a^{n} b^{n}$
$a^{- m} = (a^{- 1})^{m} = (a^{m})^{- 1}$
Order of $a$ divides order of $a^{k}$

Subgroup Theory (20+)

Intersection of subgroups is a subgroup
Union of subgroups is NOT generally a subgroup
One-step subgroup test
Two-step subgroup test
Finite subgroup test
Cyclic subgroups exist for every element
$⟨ a ⟩ = a^{n} : n \in Z$
Center $Z (G)$ is a subgroup
Centralizer $C (a)$ is a subgroup
Normalizer $N (H)$ is a subgroup
Stabilizer is a subgroup
Kernel of homomorphism is a subgroup
Image of homomorphism is a subgroup
Subgroups of cyclic groups are cyclic
Every subgroup of $Z$ is of form $n Z$
Index satisfies: $∣ G ∣ = ∣ H ∣ \cdot [G : H]$
Lattice of subgroups has properties
Join and meet of subgroups
Subgroup generated by subset $S$ : $⟨ S ⟩$
Commutator subgroup $[G, G]$ is normal
Derived series properties

Cosets and Lagrange (15+)

Lagrange’s Theorem: $∣ H ∣$ divides $∣ G ∣$
Left cosets partition the group
Right cosets partition the group
$∣ a H ∣ = ∣ H ∣$ for any coset
$a H = b H$ iff $b^{- 1} a \in H$
Number of left cosets = number of right cosets
Index $[G : H] = ∣ G ∣/∣ H ∣$
Order of element divides order of group
$a^{∣ G ∣} = e$ for all $a$ in finite group
Group of prime order is cyclic
Subgroups of index 2 are normal
$(G : K) = (G : H) (H : K)$ (tower law)
Orbit-stabilizer theorem
Burnside’s lemma (orbit counting)
Class equation

Normal Subgroups and Quotients (20+)

$H ◃ G$ iff $g H g^{- 1} = H$ for all $g$
$H ◃ G$ iff $g H = H g$ for all $g$
Kernel is always normal
Center is always normal
Intersection of normal subgroups is normal
Normal subgroup of normal subgroup is NOT necessarily normal in $G$
Quotient group $G / N$ well-defined when $N$ normal
$∣ G / N ∣ = ∣ G ∣/∣ N ∣$
Correspondence theorem (fourth isomorphism theorem)
Diamond isomorphism theorem (second isomorphism theorem)
$(HK) / K ≅ H / (H \cap K)$
If $H, K ◃ G$ , then $HK ◃ G$
Product of normal subgroups is normal
Commutator subgroup is normal
Simple groups have no nontrivial normal subgroups
Composition series properties
Jordan-Hölder theorem
Solvable group properties
Nilpotent group properties
Fitting subgroup properties

Homomorphisms (25+)

Composition of homomorphisms is a homomorphism
Identity map is a homomorphism
Inverse of isomorphism is isomorphism
$ϕ (e_{G}) = e_{H}$
$ϕ (a^{- 1}) = ϕ (a)^{- 1}$
$ϕ (a^{n}) = ϕ (a)^{n}$
Order of $ϕ (a)$ divides order of $a$
Kernel is normal subgroup
Image is subgroup
$ϕ$ injective iff $ker (ϕ) = e$
First Isomorphism Theorem: $G / ker (ϕ) ≅ Im (ϕ)$
Second Isomorphism Theorem (Diamond)
Third Isomorphism Theorem: $(G / N) / (H / N) ≅ G / H$
Fourth Isomorphism Theorem (Correspondence/Lattice)
Preimage of subgroup is subgroup
Image of subgroup is subgroup
Preimage of normal subgroup is normal
Automorphisms form a group $Aut (G)$
Inner automorphisms form a group $Inn (G)$
$Inn (G) ≅ G / Z (G)$
$Inn (G) ◃ Aut (G)$
Conjugation is automorphism
Cayley’s Theorem: $G ≅$ subgroup of $S_{G}$
Embedding theorems
Universal property of quotients

Cyclic Groups (15+)

$⟨ a ⟩ = a^{n} : n \in Z$
Every cyclic group is abelian
Cyclic groups are isomorphic to $Z$ or $Z_{n}$
Subgroups of cyclic groups are cyclic
$Z_{n}$ has exactly one subgroup of order $d$ for each $d ∣ n$
Number of generators of $Z_{n}$ is $ϕ (n)$ (Euler’s totient)
$a^{k}$ generates $⟨ a ⟩$ iff $g cd (k, ∣ a ∣) = 1$
$∣ ⟨ a ⟩ ∣ = ∣ a ∣$ (order of cyclic group = order of generator)
Fundamental theorem of cyclic groups
Every group of prime order is cyclic
Quotient of cyclic group is cyclic
Image of cyclic group is cyclic
Product of cyclic groups characterization
Chinese Remainder Theorem for groups
Classification of finite abelian groups uses cyclic groups

Permutation Groups (20+)

$S_{n}$ has order $n!$
Every permutation is product of disjoint cycles
Disjoint cycles commute
Order of permutation = lcm of cycle lengths
Every permutation is product of transpositions
Even/odd permutation well-defined
$A_{n}$ (alternating group) is normal in $S_{n}$
$∣ A_{n} ∣ = n! /2$
$A_{n}$ is simple for $n \geq 5$
Cycle type determines conjugacy class in $S_{n}$
Number of conjugates = $n! / (cycle structure)$
Cayley’s theorem embeds any group in symmetric group
$S_{3}$ is non-abelian, smallest non-abelian group
$S_{n}$ is non-abelian for $n \geq 3$
Symmetric group is complete for $n \neq = 2, 6$
Primitive and imprimitive group actions
Multiply transitive group properties
Wreath product properties
Sylow theorems apply to $S_{n}$
Simplicity proofs for $A_{n}$

Direct Products (12+)

$G \times H$ is a group
$∣ G \times H ∣ = ∣ G ∣ \cdot ∣ H ∣$
$G \times H ≅ H \times G$
$(G \times H) \times K ≅ G \times (H \times K)$
$G \times e ≅ G$
$(g_{1}, h_{1}) (g_{2}, h_{2}) = (g_{1} g_{2}, h_{1} h_{2})$
Recognition theorem for internal direct products
$G ≅ H \times K$ iff certain conditions
Fundamental theorem of finite abelian groups
Every finite abelian group is product of cyclic groups
Chinese Remainder Theorem
External vs internal direct product

Group Actions (25+)

Orbit-stabilizer theorem: $∣ G ∣ = ∣ Orb (x) ∣ \cdot ∣ Stab (x) ∣$
Orbits partition the set
Burnside’s lemma: $∣ orbits ∣ = \frac{1}{∣ G ∣} \sum_{g \in G} ∣ X^{g} ∣$
Cauchy-Frobenius lemma
Class equation: $∣ G ∣ = ∣ Z (G) ∣ + \sum [G : C (x_{i})]$
Conjugacy classes partition group
Size of conjugacy class divides $∣ G ∣$
Transitive action properties
Faithful action properties
Free action properties
Cayley’s theorem as action
Left regular representation
Right regular representation
Conjugation action
Action on cosets
Action on subgroups by conjugation
Stabilizer is a subgroup
Kernel of action is normal
Primitive vs imprimitive actions
Blocks and block systems
Fixed point theorems
Sylow theorems via group actions
Cauchy’s theorem via actions
p-group fixed point theorem
Normalizer-centralizer lemmas

Sylow Theorems (10+)

First Sylow Theorem: Sylow p-subgroups exist
Second Sylow Theorem: Sylow p-subgroups are conjugate
Third Sylow Theorem: $n_{p} \equiv 1 (mod p)$ and $n_{p} ∣∣ G ∣$
Number of Sylow p-subgroups formula
Normalizer of Sylow subgroup
Intersection of Sylow subgroups
Sylow subgroups in specific groups
Applications to group structure
Fusion in Sylow subgroups
Transfer homomorphism

Finite Group Theory (30+)

Cauchy’s theorem: If $p ∣∣ G ∣$ , then $\exists a$ with $∣ a ∣ = p$
p-groups have non-trivial center
Groups of order $p^{2}$ are abelian
Classification of groups of small order
Groups of order $pq$ classification
Simple group classification (CFSG - massive theorem!)
Feit-Thompson theorem: odd order groups are solvable
Hall subgroups
Frattini subgroup
Fitting subgroup
Socle
Commutator calculus
Nilpotent groups characterization
Solvable groups characterization
Composition series uniqueness
Chief series
Schreier refinement theorem
Transfer theorems
Burnside’s $p^{a} q^{b}$ theorem
Frobenius groups
Zassenhaus lemma
Schur-Zassenhaus theorem
Hall’s theorem
Wielandt’s theorems
Thompson subgroup
Glauberman’s Z* theorem
Character theory fundamentals
Orthogonality relations
Induced characters
Frobenius reciprocity

Abelian Groups (20+)

All subgroups of abelian group are normal
Quotient of abelian group is abelian
Homomorphic image of abelian is abelian
Fundamental Theorem of Finitely Generated Abelian Groups
Structure theorem: $G ≅ Z^{r} \times Z_{n_{1}} \times \dots \times Z_{n_{k}}$
Rank and torsion decomposition
Primary decomposition
Invariant factors
Elementary divisors
Uniqueness of decomposition
Ulm invariants (infinite case)
Divisible groups
Injective abelian groups
Torsion subgroup is subgroup
Torsion-free groups
Pure subgroups
Basic subgroups
Pontryagin duality
Character groups
Ext and Tor functors

Free Groups (15+)

Free groups exist
Universal property of free groups
Every group is quotient of free group
Presentation of groups: $⟨ S ∣ R ⟩$
Normal form in free groups
Subgroups of free groups are free (Nielsen-Schreier)
Rank of subgroup formula
Free product properties
Free product with amalgamation
HNN extensions
Van Kampen’s theorem
Word problem
Conjugacy problem
Isomorphism problem
Tietze transformations

Matrix Groups (10+)

$G L_{n} (F)$ is a group
$S L_{n} (F)$ is normal in $G L_{n}$
$O (n)$ and $SO (n)$ properties
$U (n)$ and $S U (n)$ properties
Determinant is homomorphism
Trace properties
Conjugacy in matrix groups
Exponential map for matrix groups
Lie correspondence for matrix groups
Classical groups structure

Solvable and Nilpotent Groups (15+)

Derived series
Lower central series
Upper central series
Solvable group characterization
Nilpotent group characterization
Subgroups and quotients of solvable are solvable
Subgroups and quotients of nilpotent are nilpotent
Nilpotent implies solvable
Finite p-groups are nilpotent
Nilpotent groups have normal Sylow subgroups
Fitting subgroup is nilpotent
Frattini subgroup properties
Three subgroups lemma
Commutator collection formulas
Hall-Witt identity

Representation Theory (20+)

Matrix representations
Degree of representation
Equivalent representations
Reducible vs irreducible
Maschke’s theorem
Schur’s lemma
Character of representation
Character table
Orthogonality of characters
Number of irreducible representations = number of conjugacy classes
Regular representation
Induced representations
Frobenius reciprocity
Tensor products of representations
Burnside’s theorem via characters
Artin’s theorem
Brauer’s theorems
Modular representation theory
Decomposition numbers
Cartan matrix

Miscellaneous Important Results (20+)

Schur-Zassenhaus theorem
Hall’s theorem on solvable groups
Gaschutz’s theorem
Frattini’s argument
Dedekind’s modular law
Butterfly lemma (Zassenhaus)
Three subgroup lemma
Commutator identities
P. Hall’s collection formula
Lazard correspondence
Malcev correspondence
Krull-Schmidt theorem
Jordan-Holder theorem uniqueness
Remak-Krull-Schmidt theorem
Fitting’s theorem
Baer’s theorem
Gruenberg’s theorem
Schur multiplier
Extension theory
Cohomology of groups

Explorer

Groups

Definition

The Four Axioms

Free Access to Properties

Foundational Properties (10)

Exponent/Power Laws (8)

Subgroup Theory (20+)

Cosets and Lagrange (15+)

Normal Subgroups and Quotients (20+)

Homomorphisms (25+)

Cyclic Groups (15+)

Permutation Groups (20+)

Direct Products (12+)

Group Actions (25+)

Sylow Theorems (10+)

Finite Group Theory (30+)

Abelian Groups (20+)

Free Groups (15+)

Matrix Groups (10+)

Solvable and Nilpotent Groups (15+)

Representation Theory (20+)

Miscellaneous Important Results (20+)

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