Group Theory: A Comprehensive Reference

A single-page overview of undergraduate group theory, from the axioms through the Sylow theorems and the structure of finite abelian groups. Includes key definitions, theorems, and proof sketches with a dependency diagram.

Folio Official

March 1, 2026

1. Introduction: How to Use This Article

This article provides a single-page panoramic view of undergraduate group theory—roughly one textbook's worth of algebra. For each topic, we present the key definitions, major theorems, and proof sketches in concise form.

How to use this page:

Newcomers: read the sections in order to build a coherent picture of the subject.
Review or exam preparation: jump to any section from the table of contents.
Quick theorem lookup: Section 13, “ Theorem Directory,”{} collects all the major results in one list.

The dependency diagram below shows how the main topics are interrelated.

graph TD
    A["Group Axioms and Basic Properties"] --> B["Subgroups and Generation"]
    B --> C["Cosets and Lagrange's Theorem"]
    C --> D["Normal Subgroups and Quotient Groups"]
    A --> E["Homomorphisms"]
    D --> E
    E --> F["The Isomorphism Theorems"]
    D --> F
    A --> G["Group Actions"]
    C --> G
    G --> H["Conjugacy and the Class Equation"]
    H --> I["The Sylow Theorems"]
    C --> I
    G --> I
    B --> J["Direct Products and the Structure of Finite Abelian Groups"]
    I --> J
    D --> K["Simple Groups and the Jordan-Hölder Theorem"]
    F --> K

    style A fill:#f5f5f5,stroke:#333,color:#000
    style I fill:#f5f5f5,stroke:#333,color:#000
    style J fill:#f5f5f5,stroke:#333,color:#000
    style K fill:#f5f5f5,stroke:#333,color:#000

Remark 1.

This article is designed to accompany the “ Group Theory Textbook”{} series. Links at the end of each section point to the corresponding full-length chapter for detailed exposition and complete proofs.

2. Group Axioms and Basic Properties

Definition 2 (Group).

A group is a pair

(G, \cdot)

consisting of a set

G

and a binary operation

\cdot : G \times G \to G

satisfying three axioms:

Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in G$ .
Identity element: there exists $e \in G$ such that $e \cdot a = a \cdot e = a$ for all $a \in G$ .
Inverses: for every $a \in G$ there exists $b \in G$ with $a \cdot b = b \cdot a = e$ .

Theorem 3 (Uniqueness of the identity and inverses).

In any group

G

the identity element is unique, and each element

a

has a unique inverse.

Proof.

Uniqueness of the identity: if

e

and

e^{'}

are both identity elements then

e = e \cdot e^{'} = e^{'}

. Uniqueness of inverses: if

b

and

b^{'}

are both inverses of

a

then

b = b (a b^{'}) = (ba) b^{'} = b^{'}

. □

Theorem 4 (Cancellation laws).

In a group

G

ab = a c \Rightarrow b = c

(left cancellation) and

ba = c a \Rightarrow b = c

(right cancellation).

Fundamental examples:

$(Z, +)$ : the additive group of integers. Identity is $0$ ; the inverse of $a$ is $- a$ .
$(S_{n}, \circ)$ : the symmetric group on $n$ letters (all permutations of ${1, \dots, n}$ ). $∣ S_{n} ∣ = n!$ .
$(G L_{n} (R), \cdot)$ : the general linear group (all invertible $n \times n$ real matrices).
$(D_{n}, \circ)$ : the dihedral group of the regular $n$ -gon. $∣ D_{n} ∣ = 2 n$ .
$(Z / n Z, +)$ : the additive group of integers modulo $n$ . A cyclic group of order $n$ .
$(V_{4}, \cdot)$ : the Klein four-group ${e, a, b, ab}$ with $a^{2} = b^{2} = (ab)^{2} = e$ .

Definition 5 (Abelian group).

A group

G

is called abelian (or commutative) if

ab = ba

for all

a, b \in G

Remark 6.

Z

Z / n Z

, and

V_{4}

are abelian. The groups

S_{n}

(

n \geq 3

D_{n}

(

n \geq 3

), and

G L_{n} (R)

(

n \geq 2

) are non-abelian.

For more details:

https://interconnectd.app/articles/iHZny7PRhPgi0urg8COk

3. Subgroups and Generation

Definition 7 (Subgroup).

A nonempty subset

H

of a group

G

is a subgroup of

G

H

is itself a group under the operation of

G

. We write

H \leq G

Theorem 8 (Subgroup criteria).

For a nonempty subset

H \subseteq G

, the following are equivalent:

$H \leq G$ .
For all $a, b \in H$ , $a b^{- 1} \in H$ (one-step test).
$e \in H$ , and for all $a, b \in H$ , $ab \in H$ and $a^{- 1} \in H$ (two-step test).

Definition 9 (Generated subgroup).

For

S \subseteq G

, the smallest subgroup of

G

containing

S

is denoted

⟨ S ⟩

and called the subgroup generated by

S

. If

G = ⟨ g ⟩

for some element

g

, then

G

is called a cyclic group.

Theorem 10 (Classification of cyclic groups).

Every cyclic group is isomorphic to

Z

(infinite cyclic) or to

Z / n Z

(cyclic of order

n

Definition 11 (Order of an element).

For

a \in G

, the order of

a

is the smallest positive integer

n

such that

a^{n} = e

; we write

ord (a) = n

. If no such

n

exists, we set

ord (a) = \infty

Proposition 12.

ord (a) = n

, then

a^{k} = e

if and only if

n ∣ k

4. Cosets and Lagrange's Theorem

Definition 13 (Cosets).

Let

H \leq G

and

a \in G

. The sets

a H = {ah ∣ h \in H} (left coset), H a = {ha ∣ h \in H} (right coset)

are called cosets of

H

. The number of distinct left cosets

[G : H] = ∣ G / H ∣

is called the index of

H

G

Remark 14.

The collection of all left cosets

{a H ∣ a \in G}

forms a partition of

G

: any two left cosets are either equal or disjoint, and their union is all of

G

Theorem 15 (Lagrange's theorem).

G

is a finite group and

H \leq G

, then

∣ G ∣ = [G : H] \cdot ∣ H ∣.

In particular,

∣ H ∣

divides

∣ G ∣

Proof.

Each left coset

a H

has exactly

∣ H ∣

elements (the map

h \mapsto ah

is a bijection

H \to a H

). Since the left cosets partition

G

, we have

∣ G ∣ = [G : H] \cdot ∣ H ∣

. □

Key corollaries:

Corollary 16.

The order of every element

a

of a finite group

G

divides

∣ G ∣

. In particular,

a^{∣ G ∣} = e

Corollary 17.

Every group of prime order

p

is cyclic (

G ≅ Z / p Z

Corollary 18 (Fermat's little theorem).

p

is prime and

g cd (a, p) = 1

, then

a^{p - 1} \equiv 1 (mod p)

Proof.

The group

(Z / p Z)^{\times}

has order

p - 1

, and

a mod p

is an element of this group. By the corollary above,

a^{p - 1} \equiv 1

. □

For more details:

https://interconnectd.app/articles/BN3fsSxdENo8Fr5b6xvu

5. Normal Subgroups and Quotient Groups

Definition 19 (Normal subgroup).

A subgroup

N \leq G

is called normal if

g N = N g

for every

g \in G

. We write

N ⊴ G

Proposition 20 (Equivalent characterizations of normality).

For

N \leq G

, the following are equivalent:

$N ⊴ G$ (i.e., $g N = N g$ for all $g \in G$ ).
$g N g^{- 1} = N$ for all $g \in G$ .
$g n g^{- 1} \in N$ for all $g \in G$ and $n \in N$ .
$N$ is the kernel of some homomorphism.

Definition 21 (Quotient group).

N ⊴ G

, the set of cosets

G / N = {g N ∣ g \in G}

becomes a group under the multiplication

(a N) (b N) = (ab) N .

This group is called the quotient group (or factor group) of

G

N

The canonical projection $π : G \to G / N$ defined by $g \mapsto g N$ is a surjective homomorphism with $ker π = N$ :

Remark 22.

The quotient construction is well-defined only for normal subgroups. If

H \leq G

is not normal, the product

(a H) (b H)

depends on the choice of coset representatives and does not yield a consistent group operation.

For more details:

https://interconnectd.app/articles/jmREBCJRSgX5rW5Eq2Xq

6. Homomorphisms

Definition 23 (Group homomorphism).

A map

φ : G \to H

between groups is a group homomorphism if

φ (ab) = φ (a) φ (b) for all a, b \in G .

Proposition 24 (Automatic preservation).

φ : G \to H

is a group homomorphism, then:

$φ (e_{G}) = e_{H}$ (the identity is preserved).
$φ (a^{- 1}) = φ (a)^{- 1}$ (inverses are preserved).

Proof.

(1) From

φ (e_{G}) = φ (e_{G} \cdot e_{G}) = φ (e_{G}) φ (e_{G})

, multiply both sides by

φ (e_{G})^{- 1}

to get

e_{H} = φ (e_{G})

. (2)

φ (a) φ (a^{- 1}) = φ (a a^{- 1}) = φ (e_{G}) = e_{H}

, so

φ (a^{- 1}) = φ (a)^{- 1}

. □

Definition 25 (Kernel and image).

For a homomorphism

φ : G \to H

ker φ Im φ = {g \in G ∣ φ (g) = e_{H}} (kernel) = {φ (g) ∣ g \in G} (image)

Proposition 26.

ker φ ⊴ G

(the kernel is a normal subgroup) and

Im φ \leq H

(the image is a subgroup).

Proposition 27.

φ

is injective if and only if

ker φ = {e_{G}}

Standard examples of homomorphisms:

The canonical projection $π : Z \to Z / n Z$ ; $ker π = n Z$ .
The sign homomorphism $sgn : S_{n} \to {+ 1, - 1}$ ; $ker (sgn) = A_{n}$ (the alternating group).
The determinant $det : G L_{n} (R) \to R^{*}$ ; $ker (det) = S L_{n} (R)$ (the special linear group).

For more details:

https://interconnectd.app/articles/IZaIuMjtorMoeOSmpMqO

7. The Isomorphism Theorems

graph LR
    A["First Isomorphism Theorem<br/>G/ker φ ≅ Im φ"] --> B["Second Isomorphism Theorem<br/>HN/N ≅ H/(H∩N)"]
    A --> C["Third Isomorphism Theorem<br/>(G/N)/(H/N) ≅ G/H"]
    D["Kernel and Image of a Homomorphism"] --> A
    E["Normal Subgroups and Quotient Groups"] --> A

    style A fill:#f5f5f5,stroke:#333,color:#000

Theorem 28 (First isomorphism theorem).

φ : G \to H

is a group homomorphism, then

G / ker φ ≅ Im φ .

Proof.

Define

\overset{φ}{ˉ} (g ker φ) = φ (g)

. If

g ker φ = g^{'} ker φ

, then

g^{- 1} g^{'} \in ker φ

, so

φ (g) = φ (g^{'})

(well-defined). That

\overset{φ}{ˉ}

is a homomorphism, injective (

\overset{φ}{ˉ} (g ker φ) = e_{H} \Rightarrow g \in ker φ

), and surjective onto

Im φ

is verified directly. □

Theorem 29 (Second isomorphism theorem (diamond isomorphism theorem)).

H \leq G

and

N ⊴ G

, then

H N / N ≅ H / (H \cap N) .

Proof.

The map

φ : H \to H N / N

given by

h \mapsto h N

is a surjective homomorphism with

ker φ = H \cap N

. Apply the first isomorphism theorem. □

Theorem 30 (Third isomorphism theorem).

N ⊴ G

H ⊴ G

, and

N \subseteq H

, then

(G / N) / (H / N) ≅ G / H .

Proof.

The map

φ : G / N \to G / H

defined by

g N \mapsto g H

is a surjective homomorphism with

ker φ = H / N

. Apply the first isomorphism theorem. □

For more details:

https://interconnectd.app/articles/IZaIuMjtorMoeOSmpMqO

8. Group Actions

Definition 31 (Group action).

A group

G

acts (on the left) on a set

X

if there is a map

G \times X \to X

, written

(g, x) \mapsto g \cdot x

, satisfying:

$e \cdot x = x$ for all $x \in X$ .
$(g h) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$ .

Remark 32.

A group action is equivalent to a homomorphism

ρ : G \to Bij (X)

into the group of bijections of

X

Definition 33 (Orbit and stabilizer).

When

G

acts on

X

Orb (x) Stab (x) = {g \cdot x ∣ g \in G} (the orbit of x) = {g \in G ∣ g \cdot x = x} (the stabilizer of x)

The stabilizer

Stab (x)

is a subgroup of

G

Theorem 34 (Orbit–stabilizer theorem).

∣ Orb (x) ∣ = [G : Stab (x)] = \frac{∣ G ∣}{∣ Stab ( x ) ∣}

Proof.

Show that the map

G / Stab (x) \to Orb (x)

given by

g Stab (x) \mapsto g \cdot x

is a well-defined bijection. □

Theorem 35 (Cayley's theorem).

Every group

G

is isomorphic to a subgroup of some symmetric group

S_{X}

. If

G

is finite, then

G

is isomorphic to a subgroup of

S_{∣ G ∣}

Proof.

G

acts on itself by left multiplication (

g \cdot x = gx

). The resulting homomorphism

ρ : G \to S_{G}

is injective:

ρ (g) = id

implies

gx = x

for all

x

, so

g = e

. □

Theorem 36 (Burnside's lemma).

If a finite group

G

acts on a finite set

X

, the number of orbits is

∣ X / G ∣ = \frac{1}{∣ G ∣} g \in G \sum ∣ X^{g} ∣,

where

X^{g} = {x \in X ∣ g \cdot x = x}

is the fixed-point set of

g

For more details:

https://interconnectd.app/articles/KSG2as2QqNvSXA5AfN88

9. Conjugacy and the Class Equation

Definition 37 (Conjugacy).

Two elements

a, b \in G

are conjugate if

b = g a g^{- 1}

for some

g \in G

. Conjugacy is an equivalence relation; its equivalence classes are called conjugacy classes.

$G$ acts on itself by conjugation ( $g \cdot x = gx g^{- 1}$ ). Under this action:

The orbit of $a$ is the conjugacy class $Cl (a) = {g a g^{- 1} ∣ g \in G}$ .
The stabilizer of $a$ is the centralizer $C_{G} (a) = {g \in G ∣ g a = a g}$ .

Definition 38 (Center of a group).

The center of

G

Z (G) = {z \in G ∣ z g = g z for all g \in G} = a \in G ⋂ C_{G} (a) .

Z (G) ⊴ G

Remark 39.

a \in Z (G)

if and only if

Cl (a) = {a}

(i.e., its conjugacy class is a singleton).

Theorem 40 (Class equation).

For a finite group

G

∣ G ∣ = ∣ Z (G) ∣ + i \sum [G : C_{G} (a_{i})],

where the sum runs over one representative

a_{i}

from each conjugacy class of size

\geq 2

Proof.

By the orbit–stabilizer theorem,

∣ Cl (a) ∣ = [G : C_{G} (a)]

. Since

G

is partitioned into conjugacy classes, separate the singleton classes (which form

Z (G)

) from the rest to obtain the class equation. □

Theorem 41 (Center of a

p

-group).

p

is prime and

∣ G ∣ = p^{n}

with

n \geq 1

(i.e.,

G

is a

p

-group), then

Z (G) \neq = {e}

Proof.

In the class equation,

∣ G ∣ = p^{n}

and each term

[G : C_{G} (a_{i})]

is a divisor of

∣ G ∣

greater than

1

, hence divisible by

p

. Therefore

∣ Z (G) ∣ = ∣ G ∣ - \sum_{i} [G : C_{G} (a_{i})] \equiv 0 (mod p)

. Since

e \in Z (G)

, we have

∣ Z (G) ∣ \geq p

. □

Corollary 42.

Every group of order

p^{2}

is abelian.

Proof.

We have

∣ Z (G) ∣ \in {p, p^{2}}

. If

∣ Z (G) ∣ = p^{2}

, then

G = Z (G)

and

G

is abelian. If

∣ Z (G) ∣ = p

, then

G / Z (G)

has order

p

and is therefore cyclic:

G / Z (G) = ⟨ g Z (G)⟩

. Every element of

G

can be written

g^{i} z

with

z \in Z (G)

, and a direct computation shows

ab = ba

for all

a, b \in G

—contradicting

∣ Z (G) ∣ = p

. Hence

∣ Z (G) ∣ = p^{2}

. □

For more details:

https://interconnectd.app/articles/KSG2as2QqNvSXA5AfN88

10. The Sylow Theorems

graph TD
    A["Lagrange's Theorem"] --> B["Cauchy's Theorem"]
    B --> C["Sylow's First Theorem"]
    D["Class Equation"] --> B
    E["Group Actions<br/>(Orbit--Stabilizer Theorem)"] --> C
    C --> F["Sylow's Second Theorem"]
    C --> G["Sylow's Third Theorem"]
    E --> F
    E --> G

    style C fill:#f5f5f5,stroke:#333,color:#000
    style F fill:#f5f5f5,stroke:#333,color:#000
    style G fill:#f5f5f5,stroke:#333,color:#000

Theorem 43 (Cauchy's theorem).

G

is a finite group and a prime

p

divides

∣ G ∣

, then

G

contains an element of order

p

Proof.

Let

S = {(a_{1}, \dots, a_{p}) \in G^{p} ∣ a_{1} a_{2} \dots a_{p} = e}

. The cyclic group

Z / p Z

acts on

S

by cyclic permutation of coordinates. Since

∣ S ∣ = ∣ G ∣^{p - 1}

, we have

p ∣ ∣ S ∣

. The fixed points are tuples

(a, a, \dots, a)

with

a^{p} = e

. The identity

e

gives one fixed point, but

p ∣ ∣ S ∣

forces the existence of others, yielding an element of order

p

. □

In what follows, write $∣ G ∣ = p^{n} m$ with $p ∤ m$ . A subgroup of order $p^{n}$ is called a Sylow $p$ -subgroup of $G$ . Denote the set of all Sylow $p$ -subgroups by $Syl_{p} (G)$ and their number by $n_{p} = ∣ Syl_{p} (G) ∣$ .

Theorem 44 (Sylow's first theorem).

Sylow

p

-subgroups exist:

Syl_{p} (G) \neq = \emptyset

Theorem 45 (Sylow's second theorem).

Any two Sylow

p

-subgroups of

G

are conjugate. That is, if

P, Q \in Syl_{p} (G)

, there exists

g \in G

with

Q = g P g^{- 1}

Theorem 46 (Sylow's third theorem).

n_{p} \equiv 1 (mod p)

and

n_{p} ∣ m

(where

m = ∣ G ∣/ p^{n}

Remark 47.

By Sylow's second theorem,

n_{p} = 1

if and only if the Sylow

p

-subgroup is normal in

G

Worked example:

Example 48.

Classify all groups of order

15 = 3 \times 5

By Sylow's third theorem,

n_{3} ∣ 5

and

n_{3} \equiv 1 (mod 3)

, so

n_{3} \in {1, 5}

. But

5 \neq \equiv 1 (mod 3)

, so

n_{3} = 1

. Similarly,

n_{5} ∣ 3

and

n_{5} \equiv 1 (mod 5)

, giving

n_{5} = 1

Let

P ≅ Z /3 Z

and

Q ≅ Z /5 Z

be the unique Sylow

3

- and

5

-subgroups. Both are normal, and

P \cap Q = {e}

. Therefore

G ≅ P \times Q ≅ Z /3 Z \times Z /5 Z ≅ Z /15 Z

. Every group of order

15

is cyclic.

For more details:

https://interconnectd.app/articles/ohndrkJdtEME1LzAlbdj

11. Direct Products and the Structure of Finite Abelian Groups

Definition 49 (Direct product).

The direct product of groups

G_{1}

and

G_{2}

is the group

G_{1} \times G_{2}

whose underlying set is the Cartesian product with componentwise multiplication:

(a_{1}, a_{2}) (b_{1}, b_{2}) = (a_{1} b_{1}, a_{2} b_{2}) .

Proposition 50 (Internal direct product criterion).

N_{1}, N_{2} ⊴ G

with

N_{1} \cap N_{2} = {e}

and

G = N_{1} N_{2}

, then

G ≅ N_{1} \times N_{2}

Theorem 51 (Fundamental theorem of finite abelian groups).

Every finite abelian group

G

is isomorphic to a direct product of cyclic groups of prime-power order:

G ≅ Z / p_{1}^{e_{1}} Z \times Z / p_{2}^{e_{2}} Z \times \dots \times Z / p_{k}^{e_{k}} Z,

where the

p_{i}

are primes (not necessarily distinct) and

e_{i} \geq 1

. This decomposition is unique up to reordering of the factors.

Example 52.

The abelian groups of order

12

are, up to isomorphism, exactly:

$Z /4 Z \times Z /3 Z ≅ Z /12 Z$ (cyclic).
$Z /2 Z \times Z /2 Z \times Z /3 Z ≅ Z /2 Z \times Z /6 Z$ .

Since

12 = 2^{2} \times 3

, the two types correspond to the two partitions of

2^{2}

(4)

and

(2, 2)

Remark 53.

Z / m Z \times Z / n Z ≅ Z / mn Z

if and only if

g cd (m, n) = 1

(the Chinese remainder theorem).

For more details:

https://interconnectd.app/articles/Q2ElstdoUzpMeaoYYwqk

https://interconnectd.app/articles/VacUK2RxF4r7Jkm2tcFR

12. Simple Groups and the Jordan–Hölder Theorem

Definition 54 (Simple group).

A group

G

is simple if

G \neq = {e}

and the only normal subgroups of

G

are

{e}

and

G

itself.

Example 55.

A cyclic group

Z / p Z

of prime order is simple (by Lagrange's theorem, its only subgroups are

{e}

and the whole group).

Theorem 56 (Simplicity of

A_{n}

For

n \geq 5

, the alternating group

A_{n}

is simple.

Definition 57 (Composition series).

A composition series for a group

G

is a chain of subgroups

{e} = G_{0} ⊴ G_{1} ⊴ \dots ⊴ G_{k} = G

such that each quotient

G_{i + 1} / G_{i}

is a simple group. The quotients

G_{i + 1} / G_{i}

are called the composition factors.

Theorem 58 (Jordan–Hölder theorem).

Every finite group

G \neq = {e}

possesses a composition series. Moreover, any two composition series of

G

have the same composition factors (up to reordering and isomorphism).

Remark 59.

The Jordan–Hölder theorem guarantees a form of unique “ prime factorization”{} for groups. Just as every integer factors uniquely into primes, every finite group is built from simple groups (via composition series) in an essentially unique way. The classification of finite simple groups is one of the great achievements of twentieth-century mathematics.

For more details:

https://interconnectd.app/articles/9YYJQuMjp9eetuSk2HnO

https://interconnectd.app/articles/nK6BAcdKJrfKdxOy2LIc

13. Theorem Directory

A one-line summary of the fifteen major theorems of undergraduate group theory.

Cancellation laws: $ab = a c \Rightarrow b = c$ .
Lagrange's theorem: $∣ H ∣$ divides $∣ G ∣$ .
Fermat's little theorem: if $p$ is prime and $g cd (a, p) = 1$ , then $a^{p - 1} \equiv 1 (mod p)$ .
Classification of cyclic groups: every cyclic group is isomorphic to $Z$ or $Z / n Z$ .
First isomorphism theorem: $G / ker φ ≅ Im φ$ .
Second isomorphism theorem: $H N / N ≅ H / (H \cap N)$ .
Third isomorphism theorem: $(G / N) / (H / N) ≅ G / H$ .
Cayley's theorem: every group is isomorphic to a subgroup of a symmetric group.
Orbit–stabilizer theorem: $∣ Orb (x) ∣ = [G : Stab (x)]$ .
Burnside's lemma: $∣ X / G ∣ = \frac{1}{∣ G ∣} \sum_{g} ∣ X^{g} ∣$ .
Class equation: $∣ G ∣ = ∣ Z (G) ∣ + \sum_{i} [G : C_{G} (a_{i})]$ .
Cauchy's theorem: if $p ∣ ∣ G ∣$ , then $G$ has an element of order $p$ .
Sylow's first theorem: Sylow $p$ -subgroups exist.
Sylow's second theorem: any two Sylow $p$ -subgroups are conjugate.
Sylow's third theorem: $n_{p} \equiv 1 (mod p)$ and $n_{p} ∣ [G : P]$ .

Additionally, the following structural results:

Fundamental theorem of finite abelian groups: unique decomposition into cyclic groups of prime-power order.
Jordan–Hölder theorem: composition factors are unique up to order and isomorphism.

14. Appendix: Notation and Terminology

Symbol	Name	Meaning
$∣ G ∣$	Order of $G$	Number of elements in the group $G$
$ord (a)$	Order of $a$	Smallest positive $n$ with $a^{n} = e$
$H \leq G$	Subgroup	$H$ is a subgroup of $G$
$N ⊴ G$	Normal subgroup	$g N g^{- 1} = N$ for all $g \in G$
$[G : H]$	Index	Number of cosets of $H$ in $G$
$G / N$	Quotient group	Group of cosets of $N$ in $G$
$ker φ$	Kernel	${g ∣ φ (g) = e_{H}}$
$Im φ$	Image	${φ (g) ∣ g \in G}$
$G ≅ H$	Isomorphism	There exists an isomorphism $G \to H$
$⟨ S ⟩$	Subgroup generated by $S$	Smallest subgroup containing $S$
$Z (G)$	Center	${z \in G ∣ z g = g z for all g}$
$C_{G} (a)$	Centralizer	${g \in G ∣ g a = a g}$
$Orb (x)$	Orbit	${g \cdot x ∣ g \in G}$
$Stab (x)$	Stabilizer	${g \in G ∣ g \cdot x = x}$
$S_{n}$	Symmetric group	All permutations of $n$ elements
$A_{n}$	Alternating group	Even permutations in $S_{n}$
$D_{n}$	Dihedral group	Symmetries of the regular $n$ -gon
$G L_{n} (R)$	General linear group	Invertible $n \times n$ real matrices
$S L_{n} (R)$	Special linear group	Matrices with determinant $1$
$Syl_{p} (G)$	Sylow $p$ -subgroups	Set of all subgroups of order $p^{n}$
$n_{p}$	Number of Sylow $p$ -subgroups	$∣ Syl_{p} (G) ∣$

Group Theory Algebra Summary Theorem Reference Sylow Theorems Isomorphism Theorems Group Actions

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

Group Theory: A Comprehensive Reference

Folio Official

March 1, 2026

1. Introduction: How to Use This Article

How to use this page:

Newcomers: read the sections in order to build a coherent picture of the subject.
Review or exam preparation: jump to any section from the table of contents.
Quick theorem lookup: Section 13, “ Theorem Directory,”{} collects all the major results in one list.

The dependency diagram below shows how the main topics are interrelated.

graph TD
    A["Group Axioms and Basic Properties"] --> B["Subgroups and Generation"]
    B --> C["Cosets and Lagrange's Theorem"]
    C --> D["Normal Subgroups and Quotient Groups"]
    A --> E["Homomorphisms"]
    D --> E
    E --> F["The Isomorphism Theorems"]
    D --> F
    A --> G["Group Actions"]
    C --> G
    G --> H["Conjugacy and the Class Equation"]
    H --> I["The Sylow Theorems"]
    C --> I
    G --> I
    B --> J["Direct Products and the Structure of Finite Abelian Groups"]
    I --> J
    D --> K["Simple Groups and the Jordan-Hölder Theorem"]
    F --> K

    style A fill:#f5f5f5,stroke:#333,color:#000
    style I fill:#f5f5f5,stroke:#333,color:#000
    style J fill:#f5f5f5,stroke:#333,color:#000
    style K fill:#f5f5f5,stroke:#333,color:#000

Remark 1.

2. Group Axioms and Basic Properties

Definition 2 (Group).

A group is a pair

(G, \cdot)

consisting of a set

G

and a binary operation

\cdot : G \times G \to G

satisfying three axioms:

Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c \in G$ .
Identity element: there exists $e \in G$ such that $e \cdot a = a \cdot e = a$ for all $a \in G$ .
Inverses: for every $a \in G$ there exists $b \in G$ with $a \cdot b = b \cdot a = e$ .

Theorem 3 (Uniqueness of the identity and inverses).

In any group

G

the identity element is unique, and each element

a

has a unique inverse.

Proof.

Uniqueness of the identity: if

e

and

e^{'}

are both identity elements then

e = e \cdot e^{'} = e^{'}

. Uniqueness of inverses: if

b

and

b^{'}

are both inverses of

a

then

b = b (a b^{'}) = (ba) b^{'} = b^{'}

. □

Theorem 4 (Cancellation laws).

In a group

G

ab = a c \Rightarrow b = c

(left cancellation) and

ba = c a \Rightarrow b = c

(right cancellation).

Fundamental examples:

$(Z, +)$ : the additive group of integers. Identity is $0$ ; the inverse of $a$ is $- a$ .
$(S_{n}, \circ)$ : the symmetric group on $n$ letters (all permutations of ${1, \dots, n}$ ). $∣ S_{n} ∣ = n!$ .
$(G L_{n} (R), \cdot)$ : the general linear group (all invertible $n \times n$ real matrices).
$(D_{n}, \circ)$ : the dihedral group of the regular $n$ -gon. $∣ D_{n} ∣ = 2 n$ .
$(Z / n Z, +)$ : the additive group of integers modulo $n$ . A cyclic group of order $n$ .
$(V_{4}, \cdot)$ : the Klein four-group ${e, a, b, ab}$ with $a^{2} = b^{2} = (ab)^{2} = e$ .

Definition 5 (Abelian group).

A group

G

is called abelian (or commutative) if

ab = ba

for all

a, b \in G

Remark 6.

Z

Z / n Z

, and

V_{4}

are abelian. The groups

S_{n}

(

n \geq 3

D_{n}

(

n \geq 3

), and

G L_{n} (R)

(

n \geq 2

) are non-abelian.

For more details:

https://interconnectd.app/articles/iHZny7PRhPgi0urg8COk

3. Subgroups and Generation

Definition 7 (Subgroup).

A nonempty subset

H

of a group

G

is a subgroup of

G

H

is itself a group under the operation of

G

. We write

H \leq G

Theorem 8 (Subgroup criteria).

For a nonempty subset

H \subseteq G

, the following are equivalent:

$H \leq G$ .
For all $a, b \in H$ , $a b^{- 1} \in H$ (one-step test).
$e \in H$ , and for all $a, b \in H$ , $ab \in H$ and $a^{- 1} \in H$ (two-step test).

Definition 9 (Generated subgroup).

For

S \subseteq G

, the smallest subgroup of

G

containing

S

is denoted

⟨ S ⟩

and called the subgroup generated by

S

. If

G = ⟨ g ⟩

for some element

g

, then

G

is called a cyclic group.

Theorem 10 (Classification of cyclic groups).

Every cyclic group is isomorphic to

Z

(infinite cyclic) or to

Z / n Z

(cyclic of order

n

Definition 11 (Order of an element).

For

a \in G

, the order of

a

is the smallest positive integer

n

such that

a^{n} = e

; we write

ord (a) = n

. If no such

n

exists, we set

ord (a) = \infty

Proposition 12.

ord (a) = n

, then

a^{k} = e

if and only if

n ∣ k

4. Cosets and Lagrange's Theorem

Definition 13 (Cosets).

Let

H \leq G

and

a \in G

. The sets

a H = {ah ∣ h \in H} (left coset), H a = {ha ∣ h \in H} (right coset)

are called cosets of

H

. The number of distinct left cosets

[G : H] = ∣ G / H ∣

is called the index of

H

G

Remark 14.

The collection of all left cosets

{a H ∣ a \in G}

forms a partition of

G

: any two left cosets are either equal or disjoint, and their union is all of

G

Theorem 15 (Lagrange's theorem).

G

is a finite group and

H \leq G

, then

∣ G ∣ = [G : H] \cdot ∣ H ∣.

In particular,

∣ H ∣

divides

∣ G ∣

Proof.

Each left coset

a H

has exactly

∣ H ∣

elements (the map

h \mapsto ah

is a bijection

H \to a H

). Since the left cosets partition

G

, we have

∣ G ∣ = [G : H] \cdot ∣ H ∣

. □

Key corollaries:

Corollary 16.

The order of every element

a

of a finite group

G

divides

∣ G ∣

. In particular,

a^{∣ G ∣} = e

Corollary 17.

Every group of prime order

p

is cyclic (

G ≅ Z / p Z

Corollary 18 (Fermat's little theorem).

p

is prime and

g cd (a, p) = 1

, then

a^{p - 1} \equiv 1 (mod p)

Proof.

The group

(Z / p Z)^{\times}

has order

p - 1

, and

a mod p

is an element of this group. By the corollary above,

a^{p - 1} \equiv 1

. □

For more details:

https://interconnectd.app/articles/BN3fsSxdENo8Fr5b6xvu

5. Normal Subgroups and Quotient Groups

Definition 19 (Normal subgroup).

A subgroup

N \leq G

is called normal if

g N = N g

for every

g \in G

. We write

N ⊴ G

Proposition 20 (Equivalent characterizations of normality).

For

N \leq G

, the following are equivalent:

$N ⊴ G$ (i.e., $g N = N g$ for all $g \in G$ ).
$g N g^{- 1} = N$ for all $g \in G$ .
$g n g^{- 1} \in N$ for all $g \in G$ and $n \in N$ .
$N$ is the kernel of some homomorphism.

Definition 21 (Quotient group).

N ⊴ G

, the set of cosets

G / N = {g N ∣ g \in G}

becomes a group under the multiplication

(a N) (b N) = (ab) N .

This group is called the quotient group (or factor group) of

G

N

The canonical projection $π : G \to G / N$ defined by $g \mapsto g N$ is a surjective homomorphism with $ker π = N$ :

Remark 22.

The quotient construction is well-defined only for normal subgroups. If

H \leq G

is not normal, the product

(a H) (b H)

depends on the choice of coset representatives and does not yield a consistent group operation.

For more details:

https://interconnectd.app/articles/jmREBCJRSgX5rW5Eq2Xq

6. Homomorphisms

Definition 23 (Group homomorphism).

A map

φ : G \to H

between groups is a group homomorphism if

φ (ab) = φ (a) φ (b) for all a, b \in G .

Proposition 24 (Automatic preservation).

φ : G \to H

is a group homomorphism, then:

$φ (e_{G}) = e_{H}$ (the identity is preserved).
$φ (a^{- 1}) = φ (a)^{- 1}$ (inverses are preserved).

Proof.

(1) From

φ (e_{G}) = φ (e_{G} \cdot e_{G}) = φ (e_{G}) φ (e_{G})

, multiply both sides by

φ (e_{G})^{- 1}

to get

e_{H} = φ (e_{G})

. (2)

φ (a) φ (a^{- 1}) = φ (a a^{- 1}) = φ (e_{G}) = e_{H}

, so

φ (a^{- 1}) = φ (a)^{- 1}

. □

Definition 25 (Kernel and image).

For a homomorphism

φ : G \to H

ker φ Im φ = {g \in G ∣ φ (g) = e_{H}} (kernel) = {φ (g) ∣ g \in G} (image)

Proposition 26.

ker φ ⊴ G

(the kernel is a normal subgroup) and

Im φ \leq H

(the image is a subgroup).

Proposition 27.

φ

is injective if and only if

ker φ = {e_{G}}

Standard examples of homomorphisms:

The canonical projection $π : Z \to Z / n Z$ ; $ker π = n Z$ .
The sign homomorphism $sgn : S_{n} \to {+ 1, - 1}$ ; $ker (sgn) = A_{n}$ (the alternating group).
The determinant $det : G L_{n} (R) \to R^{*}$ ; $ker (det) = S L_{n} (R)$ (the special linear group).

For more details:

https://interconnectd.app/articles/IZaIuMjtorMoeOSmpMqO

7. The Isomorphism Theorems

graph LR
    A["First Isomorphism Theorem<br/>G/ker φ ≅ Im φ"] --> B["Second Isomorphism Theorem<br/>HN/N ≅ H/(H∩N)"]
    A --> C["Third Isomorphism Theorem<br/>(G/N)/(H/N) ≅ G/H"]
    D["Kernel and Image of a Homomorphism"] --> A
    E["Normal Subgroups and Quotient Groups"] --> A

    style A fill:#f5f5f5,stroke:#333,color:#000

Theorem 28 (First isomorphism theorem).

φ : G \to H

is a group homomorphism, then

G / ker φ ≅ Im φ .

Proof.

Define

\overset{φ}{ˉ} (g ker φ) = φ (g)

. If

g ker φ = g^{'} ker φ

, then

g^{- 1} g^{'} \in ker φ

, so

φ (g) = φ (g^{'})

(well-defined). That

\overset{φ}{ˉ}

is a homomorphism, injective (

\overset{φ}{ˉ} (g ker φ) = e_{H} \Rightarrow g \in ker φ

), and surjective onto

Im φ

is verified directly. □

Theorem 29 (Second isomorphism theorem (diamond isomorphism theorem)).

H \leq G

and

N ⊴ G

, then

H N / N ≅ H / (H \cap N) .

Proof.

The map

φ : H \to H N / N

given by

h \mapsto h N

is a surjective homomorphism with

ker φ = H \cap N

. Apply the first isomorphism theorem. □

Theorem 30 (Third isomorphism theorem).

N ⊴ G

H ⊴ G

, and

N \subseteq H

, then

(G / N) / (H / N) ≅ G / H .

Proof.

The map

φ : G / N \to G / H

defined by

g N \mapsto g H

is a surjective homomorphism with

ker φ = H / N

. Apply the first isomorphism theorem. □

For more details:

https://interconnectd.app/articles/IZaIuMjtorMoeOSmpMqO

8. Group Actions

Definition 31 (Group action).

A group

G

acts (on the left) on a set

X

if there is a map

G \times X \to X

, written

(g, x) \mapsto g \cdot x

, satisfying:

$e \cdot x = x$ for all $x \in X$ .
$(g h) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$ .

Remark 32.

A group action is equivalent to a homomorphism

ρ : G \to Bij (X)

into the group of bijections of

X

Definition 33 (Orbit and stabilizer).

When

G

acts on

X

Orb (x) Stab (x) = {g \cdot x ∣ g \in G} (the orbit of x) = {g \in G ∣ g \cdot x = x} (the stabilizer of x)

The stabilizer

Stab (x)

is a subgroup of

G

Theorem 34 (Orbit–stabilizer theorem).

∣ Orb (x) ∣ = [G : Stab (x)] = \frac{∣ G ∣}{∣ Stab ( x ) ∣}

Proof.

Show that the map

G / Stab (x) \to Orb (x)

given by

g Stab (x) \mapsto g \cdot x

is a well-defined bijection. □

Theorem 35 (Cayley's theorem).

Every group

G

is isomorphic to a subgroup of some symmetric group

S_{X}

. If

G

is finite, then

G

is isomorphic to a subgroup of

S_{∣ G ∣}

Proof.

G

acts on itself by left multiplication (

g \cdot x = gx

). The resulting homomorphism

ρ : G \to S_{G}

is injective:

ρ (g) = id

implies

gx = x

for all

x

, so

g = e

. □

Theorem 36 (Burnside's lemma).

If a finite group

G

acts on a finite set

X

, the number of orbits is

∣ X / G ∣ = \frac{1}{∣ G ∣} g \in G \sum ∣ X^{g} ∣,

where

X^{g} = {x \in X ∣ g \cdot x = x}

is the fixed-point set of

g

For more details:

https://interconnectd.app/articles/KSG2as2QqNvSXA5AfN88

9. Conjugacy and the Class Equation

Definition 37 (Conjugacy).

Two elements

a, b \in G

are conjugate if

b = g a g^{- 1}

for some

g \in G

. Conjugacy is an equivalence relation; its equivalence classes are called conjugacy classes.

$G$ acts on itself by conjugation ( $g \cdot x = gx g^{- 1}$ ). Under this action:

The orbit of $a$ is the conjugacy class $Cl (a) = {g a g^{- 1} ∣ g \in G}$ .
The stabilizer of $a$ is the centralizer $C_{G} (a) = {g \in G ∣ g a = a g}$ .

Definition 38 (Center of a group).

The center of

G

Z (G) = {z \in G ∣ z g = g z for all g \in G} = a \in G ⋂ C_{G} (a) .

Z (G) ⊴ G

Remark 39.

a \in Z (G)

if and only if

Cl (a) = {a}

(i.e., its conjugacy class is a singleton).

Theorem 40 (Class equation).

For a finite group

G

∣ G ∣ = ∣ Z (G) ∣ + i \sum [G : C_{G} (a_{i})],

where the sum runs over one representative

a_{i}

from each conjugacy class of size

\geq 2

Proof.

By the orbit–stabilizer theorem,

∣ Cl (a) ∣ = [G : C_{G} (a)]

. Since

G

is partitioned into conjugacy classes, separate the singleton classes (which form

Z (G)

) from the rest to obtain the class equation. □

Theorem 41 (Center of a

p

-group).

p

is prime and

∣ G ∣ = p^{n}

with

n \geq 1

(i.e.,

G

is a

p

-group), then

Z (G) \neq = {e}

Proof.

In the class equation,

∣ G ∣ = p^{n}

and each term

[G : C_{G} (a_{i})]

is a divisor of

∣ G ∣

greater than

1

, hence divisible by

p

. Therefore

∣ Z (G) ∣ = ∣ G ∣ - \sum_{i} [G : C_{G} (a_{i})] \equiv 0 (mod p)

. Since

e \in Z (G)

, we have

∣ Z (G) ∣ \geq p

. □

Corollary 42.

Every group of order

p^{2}

is abelian.

Proof.

We have

∣ Z (G) ∣ \in {p, p^{2}}

. If

∣ Z (G) ∣ = p^{2}

, then

G = Z (G)

and

G

is abelian. If

∣ Z (G) ∣ = p

, then

G / Z (G)

has order

p

and is therefore cyclic:

G / Z (G) = ⟨ g Z (G)⟩

. Every element of

G

can be written

g^{i} z

with

z \in Z (G)

, and a direct computation shows

ab = ba

for all

a, b \in G

—contradicting

∣ Z (G) ∣ = p

. Hence

∣ Z (G) ∣ = p^{2}

. □

For more details:

https://interconnectd.app/articles/KSG2as2QqNvSXA5AfN88

10. The Sylow Theorems

graph TD
    A["Lagrange's Theorem"] --> B["Cauchy's Theorem"]
    B --> C["Sylow's First Theorem"]
    D["Class Equation"] --> B
    E["Group Actions<br/>(Orbit--Stabilizer Theorem)"] --> C
    C --> F["Sylow's Second Theorem"]
    C --> G["Sylow's Third Theorem"]
    E --> F
    E --> G

    style C fill:#f5f5f5,stroke:#333,color:#000
    style F fill:#f5f5f5,stroke:#333,color:#000
    style G fill:#f5f5f5,stroke:#333,color:#000

Theorem 43 (Cauchy's theorem).

G

is a finite group and a prime

p

divides

∣ G ∣

, then

G

contains an element of order

p

Proof.

Let

S = {(a_{1}, \dots, a_{p}) \in G^{p} ∣ a_{1} a_{2} \dots a_{p} = e}

. The cyclic group

Z / p Z

acts on

S

by cyclic permutation of coordinates. Since

∣ S ∣ = ∣ G ∣^{p - 1}

, we have

p ∣ ∣ S ∣

. The fixed points are tuples

(a, a, \dots, a)

with

a^{p} = e

. The identity

e

gives one fixed point, but

p ∣ ∣ S ∣

forces the existence of others, yielding an element of order

p

. □

Theorem 44 (Sylow's first theorem).

Sylow

p

-subgroups exist:

Syl_{p} (G) \neq = \emptyset

Theorem 45 (Sylow's second theorem).

Any two Sylow

p

-subgroups of

G

are conjugate. That is, if

P, Q \in Syl_{p} (G)

, there exists

g \in G

with

Q = g P g^{- 1}

Theorem 46 (Sylow's third theorem).

n_{p} \equiv 1 (mod p)

and

n_{p} ∣ m

(where

m = ∣ G ∣/ p^{n}

Remark 47.

By Sylow's second theorem,

n_{p} = 1

if and only if the Sylow

p

-subgroup is normal in

G

Worked example:

Example 48.

Classify all groups of order

15 = 3 \times 5

By Sylow's third theorem,

n_{3} ∣ 5

and

n_{3} \equiv 1 (mod 3)

, so

n_{3} \in {1, 5}

. But

5 \neq \equiv 1 (mod 3)

, so

n_{3} = 1

. Similarly,

n_{5} ∣ 3

and

n_{5} \equiv 1 (mod 5)

, giving

n_{5} = 1

Let

P ≅ Z /3 Z

and

Q ≅ Z /5 Z

be the unique Sylow

3

- and

5

-subgroups. Both are normal, and

P \cap Q = {e}

. Therefore

G ≅ P \times Q ≅ Z /3 Z \times Z /5 Z ≅ Z /15 Z

. Every group of order

15

is cyclic.

For more details:

https://interconnectd.app/articles/ohndrkJdtEME1LzAlbdj

11. Direct Products and the Structure of Finite Abelian Groups

Definition 49 (Direct product).

The direct product of groups

G_{1}

and

G_{2}

is the group

G_{1} \times G_{2}

whose underlying set is the Cartesian product with componentwise multiplication:

(a_{1}, a_{2}) (b_{1}, b_{2}) = (a_{1} b_{1}, a_{2} b_{2}) .

Proposition 50 (Internal direct product criterion).

N_{1}, N_{2} ⊴ G

with

N_{1} \cap N_{2} = {e}

and

G = N_{1} N_{2}

, then

G ≅ N_{1} \times N_{2}

Theorem 51 (Fundamental theorem of finite abelian groups).

Every finite abelian group

G

is isomorphic to a direct product of cyclic groups of prime-power order:

G ≅ Z / p_{1}^{e_{1}} Z \times Z / p_{2}^{e_{2}} Z \times \dots \times Z / p_{k}^{e_{k}} Z,

where the

p_{i}

are primes (not necessarily distinct) and

e_{i} \geq 1

. This decomposition is unique up to reordering of the factors.

Example 52.

The abelian groups of order

12

are, up to isomorphism, exactly:

$Z /4 Z \times Z /3 Z ≅ Z /12 Z$ (cyclic).
$Z /2 Z \times Z /2 Z \times Z /3 Z ≅ Z /2 Z \times Z /6 Z$ .

Since

12 = 2^{2} \times 3

, the two types correspond to the two partitions of

2^{2}

(4)

and

(2, 2)

Remark 53.

Z / m Z \times Z / n Z ≅ Z / mn Z

if and only if

g cd (m, n) = 1

(the Chinese remainder theorem).

For more details:

https://interconnectd.app/articles/Q2ElstdoUzpMeaoYYwqk

https://interconnectd.app/articles/VacUK2RxF4r7Jkm2tcFR

12. Simple Groups and the Jordan–Hölder Theorem

Definition 54 (Simple group).

A group

G

is simple if

G \neq = {e}

and the only normal subgroups of

G

are

{e}

and

G

itself.

Example 55.

A cyclic group

Z / p Z

of prime order is simple (by Lagrange's theorem, its only subgroups are

{e}

and the whole group).

Theorem 56 (Simplicity of

A_{n}

For

n \geq 5

, the alternating group

A_{n}

is simple.

Definition 57 (Composition series).

A composition series for a group

G

is a chain of subgroups

{e} = G_{0} ⊴ G_{1} ⊴ \dots ⊴ G_{k} = G

such that each quotient

G_{i + 1} / G_{i}

is a simple group. The quotients

G_{i + 1} / G_{i}

are called the composition factors.

Theorem 58 (Jordan–Hölder theorem).

Every finite group

G \neq = {e}

possesses a composition series. Moreover, any two composition series of

G

have the same composition factors (up to reordering and isomorphism).

Remark 59.

For more details:

https://interconnectd.app/articles/9YYJQuMjp9eetuSk2HnO

https://interconnectd.app/articles/nK6BAcdKJrfKdxOy2LIc

13. Theorem Directory

A one-line summary of the fifteen major theorems of undergraduate group theory.

Cancellation laws: $ab = a c \Rightarrow b = c$ .
Lagrange's theorem: $∣ H ∣$ divides $∣ G ∣$ .
Fermat's little theorem: if $p$ is prime and $g cd (a, p) = 1$ , then $a^{p - 1} \equiv 1 (mod p)$ .
Classification of cyclic groups: every cyclic group is isomorphic to $Z$ or $Z / n Z$ .
First isomorphism theorem: $G / ker φ ≅ Im φ$ .
Second isomorphism theorem: $H N / N ≅ H / (H \cap N)$ .
Third isomorphism theorem: $(G / N) / (H / N) ≅ G / H$ .
Cayley's theorem: every group is isomorphic to a subgroup of a symmetric group.
Orbit–stabilizer theorem: $∣ Orb (x) ∣ = [G : Stab (x)]$ .
Burnside's lemma: $∣ X / G ∣ = \frac{1}{∣ G ∣} \sum_{g} ∣ X^{g} ∣$ .
Class equation: $∣ G ∣ = ∣ Z (G) ∣ + \sum_{i} [G : C_{G} (a_{i})]$ .
Cauchy's theorem: if $p ∣ ∣ G ∣$ , then $G$ has an element of order $p$ .
Sylow's first theorem: Sylow $p$ -subgroups exist.
Sylow's second theorem: any two Sylow $p$ -subgroups are conjugate.
Sylow's third theorem: $n_{p} \equiv 1 (mod p)$ and $n_{p} ∣ [G : P]$ .

Additionally, the following structural results:

Fundamental theorem of finite abelian groups: unique decomposition into cyclic groups of prime-power order.
Jordan–Hölder theorem: composition factors are unique up to order and isomorphism.

14. Appendix: Notation and Terminology

Symbol	Name	Meaning
$∣ G ∣$	Order of $G$	Number of elements in the group $G$
$ord (a)$	Order of $a$	Smallest positive $n$ with $a^{n} = e$
$H \leq G$	Subgroup	$H$ is a subgroup of $G$
$N ⊴ G$	Normal subgroup	$g N g^{- 1} = N$ for all $g \in G$
$[G : H]$	Index	Number of cosets of $H$ in $G$
$G / N$	Quotient group	Group of cosets of $N$ in $G$
$ker φ$	Kernel	${g ∣ φ (g) = e_{H}}$
$Im φ$	Image	${φ (g) ∣ g \in G}$
$G ≅ H$	Isomorphism	There exists an isomorphism $G \to H$
$⟨ S ⟩$	Subgroup generated by $S$	Smallest subgroup containing $S$
$Z (G)$	Center	${z \in G ∣ z g = g z for all g}$
$C_{G} (a)$	Centralizer	${g \in G ∣ g a = a g}$
$Orb (x)$	Orbit	${g \cdot x ∣ g \in G}$
$Stab (x)$	Stabilizer	${g \in G ∣ g \cdot x = x}$
$S_{n}$	Symmetric group	All permutations of $n$ elements
$A_{n}$	Alternating group	Even permutations in $S_{n}$
$D_{n}$	Dihedral group	Symmetries of the regular $n$ -gon
$G L_{n} (R)$	General linear group	Invertible $n \times n$ real matrices
$S L_{n} (R)$	Special linear group	Matrices with determinant $1$
$Syl_{p} (G)$	Sylow $p$ -subgroups	Set of all subgroups of order $p^{n}$
$n_{p}$	Number of Sylow $p$ -subgroups	$∣ Syl_{p} (G) ∣$

Group Theory Algebra Summary Theorem Reference Sylow Theorems Isomorphism Theorems Group Actions

Folio Official

Mathematics "between the lines" — exploring the intuition textbooks leave out, written in LaTeX on Folio.

1 followers·107 articles

1. Introduction: How to Use This Article

2. Group Axioms and Basic Properties

3. Subgroups and Generation

4. Cosets and Lagrange's Theorem

5. Normal Subgroups and Quotient Groups

6. Homomorphisms

7. The Isomorphism Theorems

8. Group Actions

9. Conjugacy and the Class Equation

10. The Sylow Theorems

11. Direct Products and the Structure of Finite Abelian Groups

12. Simple Groups and the Jordan–Hölder Theorem

13. Theorem Directory

14. Appendix: Notation and Terminology

Share your expertise with the world

More from Folio Official

The Homomorphism Theorems

Group Actions and the Class Equation

Cosets and Lagrange's Theorem

Direct and Semidirect Products

1. Introduction: How to Use This Article

2. Group Axioms and Basic Properties

3. Subgroups and Generation

4. Cosets and Lagrange's Theorem

5. Normal Subgroups and Quotient Groups

6. Homomorphisms

7. The Isomorphism Theorems

8. Group Actions

9. Conjugacy and the Class Equation

10. The Sylow Theorems

11. Direct Products and the Structure of Finite Abelian Groups

12. Simple Groups and the Jordan–Hölder Theorem

13. Theorem Directory

14. Appendix: Notation and Terminology

Share your expertise with the world

More from Folio Official

The Homomorphism Theorems

Group Actions and the Class Equation

Cosets and Lagrange's Theorem

Direct and Semidirect Products