Classification of Groups: Extensive Taxonomy

I. Finite Groups

A. Abelian Finite Groups

• Cyclic groups $\mathbb{Z}/n\mathbb{Z}$
• Elementary abelian groups $(\mathbb{Z}/p\mathbb{Z})^n$
• General finite abelian groups — classified as direct products of cyclic groups of prime-power order: $G \cong \mathbb{Z}/p_1^{a_1}\mathbb{Z} \times \mathbb{Z}/p_2^{a_2}\mathbb{Z} \times \cdots \times \mathbb{Z}/p_k^{a_k}\mathbb{Z}$

B. Families of Finite Non-Abelian Groups

Permutation-Derived

• Symmetric groups $S_n$
• Alternating groups $A_n$
• Generalized symmetric groups $\mathbb{Z}_m \wr S_n$

Reflection and Rotation Groups

• Dihedral groups $D_n$ — symmetries of regular $n$-gon, order $2n$
• Dicyclic (binary dihedral) groups $\mathrm{Dic}_n$
• Finite Coxeter groups:
  • Types $A_n, B_n, C_n, D_n$
  • Exceptional types $E_6, E_7, E_8, F_4$
  • Non-crystallographic $H_3, H_4, I_2(m)$
• Finite reflection groups (crystallographic and non-crystallographic)

Extensions of Cyclic Groups

• Metacyclic groups — extension of cyclic by cyclic
• Semidihedral groups $\mathrm{SD}_{2^n}$
• Modular $p$-groups $M_{p^n}$
• Generalized quaternion groups $Q_{2^n}$

$p$-Groups

• Extraspecial $p$-groups — $Z(G) = G' = \Phi(G)$ of order $p$
• Groups of nilpotency class 2
• Regular $p$-groups
• Powerful $p$-groups

Matrix Groups over Finite Fields $\mathbb{F}_q$

| Notation | Name | |:--------:|:-----| | $\mathrm{GL}(n, q)$ | General linear group | | $\mathrm{SL}(n, q)$ | Special linear group | | $\mathrm{PGL}(n, q)$ | Projective general linear | | $\mathrm{PSL}(n, q)$ | Projective special linear | | $\mathrm{O}^\pm(n, q)$ | Orthogonal groups | | $\mathrm{SO}(n, q)$ | Special orthogonal | | $\mathrm{Sp}(2n, q)$ | Symplectic group | | $\mathrm{U}(n, q)$ | Unitary group | | $\mathrm{SU}(n, q)$ | Special unitary | | $\mathrm{PSU}(n, q)$ | Projective special unitary |

C. Finite Simple Groups (Classification Theorem)

The classification of finite simple groups (CFSG) states that every finite simple group belongs to one of the following families:

1. Cyclic of Prime Order

$\mathbb{Z}/p\mathbb{Z}, \quad p \text{ prime}$

2. Alternating Groups

$A_n, \quad n \geq 5$

3. Groups of Lie Type

Classical Chevalley Groups:

| Series | Groups | |:------:|:-------| | $A_n$ | $\mathrm{PSL}(n+1, q)$ | | $B_n$ | $\mathrm{P\Omega}(2n+1, q)$ | | $C_n$ | $\mathrm{PSp}(2n, q)$ | | $D_n$ | $\mathrm{P\Omega}^+(2n, q)$ |

Exceptional Chevalley Groups:

$G_2(q), \quad F_4(q), \quad E_6(q), \quad E_7(q), \quad E_8(q)$

Twisted (Steinberg) Groups:

${}^2A_n(q), \quad {}^2D_n(q), \quad {}^2E_6(q), \quad {}^3D_4(q)$

Suzuki and Ree Groups:

${}^2B_2(2^{2n+1}), \quad {}^2G_2(3^{2n+1}), \quad {}^2F_4(2^{2n+1})$

Tits Group:

${}^2F_4(2)'$

4. Sporadic Groups (26 Groups)

Mathieu Groups: $M_{11}, \quad M_{12}, \quad M_{22}, \quad M_{23}, \quad M_{24}$

Leech Lattice Family: $\mathrm{Co}_1, \quad \mathrm{Co}_2, \quad \mathrm{Co}_3, \quad \mathrm{McL}, \quad \mathrm{HS}, \quad \mathrm{Suz}$

Fischer Groups: $\mathrm{Fi}_{22}, \quad \mathrm{Fi}_{23}, \quad \mathrm{Fi}_{24}'$

Monster Family: $\mathbb{M} \text{ (Monster)}, \quad \mathbb{B} \text{ (Baby Monster)}$

Pariahs (not involved in the Monster): $J_1, \quad J_3, \quad J_4, \quad \mathrm{Ru}, \quad \mathrm{O'N}, \quad \mathrm{Ly}$

Others: $J_2 \text{ (Hall-Janko)}, \quad \mathrm{He}, \quad \mathrm{HN}, \quad \mathrm{Th}$

D. Structural Classes of Finite Groups

| Class | Definition | |:------|:-----------| | Simple | No proper non-trivial normal subgroups | | Quasisimple | Perfect central extension of a simple group | | Almost simple | $S \leq G \leq \mathrm{Aut}(S)$ for simple $S$ | | Solvable | Derived series terminates: $G^{(n)} = 1$ | | Supersolvable | Normal series with cyclic factors | | Nilpotent | Lower central series terminates: $\gamma_n(G) = 1$ | | Frobenius | Transitive, point stabilizers intersect trivially | | Doubly transitive | Transitive on ordered pairs | | Primitive | No non-trivial block systems |

II. Infinite Discrete Groups

A. Abelian

• Free abelian groups $\mathbb{Z}^n$
• The integers $\mathbb{Z}$
• Rationals $(\mathbb{Q}, +)$
• Quotient $\mathbb{Q}/\mathbb{Z}$
• Prüfer $p$-groups $\mathbb{Z}(p^\infty) = \varinjlim \mathbb{Z}/p^n\mathbb{Z}$
• Divisible groups — $nG = G$ for all $n$
• Torsion-free abelian groups (classification incomplete beyond rank 1)

B. Free and Combinatorial

• Free groups $F_n$ — generators $\{x_1, \ldots, x_n\}$, no relations
• Free products $G * H$
• Amalgamated free products $G *_A H$
• HNN extensions $G *_{\phi}$
• Graph products of groups
• Right-angled Artin groups (RAAGs) — defined by commutation graph
• Right-angled Coxeter groups

C. Coxeter and Reflection Groups

A Coxeter group is defined by generators $s_1, \ldots, s_n$ and relations: $(s_i s_j)^{m_{ij}} = 1, \quad m_{ii} = 1, \quad m_{ij} \geq 2 \text{ for } i \neq j$

Affine Coxeter Groups: $\tilde{A}_n, \quad \tilde{B}_n, \quad \tilde{C}_n, \quad \tilde{D}_n, \quad \tilde{E}_6, \quad \tilde{E}_7, \quad \tilde{E}_8, \quad \tilde{F}_4, \quad \tilde{G}_2$

Others:

• Hyperbolic Coxeter groups
• Artin–Tits groups (braid-like generalizations)

D. Braid and Mapping Class Groups

• Braid group $B_n$ — generators $\sigma_1, \ldots, \sigma_{n-1}$ with relations: $\sigma_i \sigma_j = \sigma_j \sigma_i \quad (|i-j| \geq 2)$ $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$
• Pure braid group $P_n = \ker(B_n \to S_n)$
• Surface braid groups
• Mapping class groups $\mathrm{Mod}(\Sigma_{g,n})$
• Torelli group $\mathcal{I}_g = \ker(\mathrm{Mod}(\Sigma_g) \to \mathrm{Sp}(2g, \mathbb{Z}))$
• Outer automorphism group $\mathrm{Out}(F_n) = \mathrm{Aut}(F_n)/\mathrm{Inn}(F_n)$

E. Geometric and Hyperbolic Groups

| Class | Characterization | |:------|:-----------------| | Hyperbolic (Gromov) | $\delta$-thin triangles in Cayley graph | | CAT(0) | Acts properly on CAT(0) space | | Automatic | Admits regular language of normal forms | | Biautomatic | Automatic with two-sided fellow traveler property | | Small cancellation | Defined by $C'(\lambda)$, $T(q)$ conditions | | One-relator | $\langle S \mid r \rangle$ | | Limit groups | Fully residually free |

F. Lattices and Arithmetic Groups

• $\mathrm{SL}(n, \mathbb{Z})$, $\mathrm{Sp}(2n, \mathbb{Z})$
• Arithmetic lattices in semisimple Lie groups
• Non-arithmetic lattices (e.g., in $\mathrm{SO}(n,1)$, $\mathrm{SU}(n,1)$)
• $S$-arithmetic groups
• Congruence subgroups $\Gamma(N) = \ker(\mathrm{SL}(n,\mathbb{Z}) \to \mathrm{SL}(n, \mathbb{Z}/N\mathbb{Z}))$
• Bianchi groups $\mathrm{PSL}(2, \mathcal{O}_K)$ for imaginary quadratic $K$
• Hilbert modular groups
• Siegel modular groups $\mathrm{Sp}(2g, \mathbb{Z})$

G. Solvable and Nilpotent Infinite Groups

• Polycyclic groups — subnormal series with cyclic factors
• Polycyclic-by-finite groups
• Finitely generated nilpotent groups
• Solvable Baumslag–Solitar groups $\mathrm{BS}(1, n) = \langle a, b \mid bab^{-1} = a^n \rangle$
• Lamplighter groups $\mathbb{Z}_m \wr \mathbb{Z}$
• Metabelian groups — $G'' = 1$

H. Groups with Finiteness Conditions

| Property | Definition | |:---------|:-----------| | Residually finite | $\bigcap_{[G:H]<\infty} H = \{1\}$ | | Hopfian | Every surjective endomorphism is injective | | co-Hopfian | Every injective endomorphism is surjective | | Type $F_n$ | Admits $K(G,1)$ with finite $n$-skeleton | | Type $FP_n$ | $\mathbb{Z}$ has projective resolution finite through degree $n$ | | Amenable | Admits invariant mean | | Property (T) | Kazhdan's property — trivial rep. is isolated | | Sofic | Approximable by finite symmetric groups | | Hyperlinear | Approximable by unitary groups |

I. Pathological and Exotic Discrete Groups

• Tarski monsters — infinite simple, all proper subgroups cyclic of prime order $p$
• Burnside groups $B(m,n) = F_m / \langle w^n \rangle$ — finitely generated of exponent $n$
• Grigorchuk group — first example of intermediate growth: $e^{n^\alpha} \prec |B_n| \prec e^{n^\beta}, \quad 0 < \alpha < \beta < 1$
• Thompson's groups $F$, $T$, $V$
• Higman's group — first finitely presented infinite simple group
• Houghton groups $H_n$
• Groups of intermediate growth

III. Topological Groups

A. General Topological Groups

• Locally compact groups
• Compact groups
• Totally disconnected locally compact (t.d.l.c.) groups
• Profinite groups — inverse limits of finite groups: $G = \varprojlim G_i, \quad |G_i| < \infty$
• Pro-$p$ groups — inverse limits of finite $p$-groups
• Prodiscrete groups

B. Specific Profinite Groups

• $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$ — profinite completion of $\mathbb{Z}$
• $\mathbb{Z}_p$ — ring of $p$-adic integers
• Galois groups $\mathrm{Gal}(\bar{K}/K)$
• Absolute Galois group $G_{\mathbb{Q}} = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
• Profinite completions $\hat{G}$ of discrete groups
• Free profinite groups

C. $p$-adic and Adelic Groups

• $\mathrm{GL}(n, \mathbb{Q}_p)$, $\mathrm{SL}(n, \mathbb{Q}_p)$
• $p$-adic Lie groups
• Adelic groups $G(\mathbb{A})$ where $\mathbb{A} = \mathbb{R} \times \prod'_p \mathbb{Q}_p$
• $\mathrm{GL}(n, \mathbb{A})$, $\mathrm{SL}(n, \mathbb{A})$

IV. Lie Groups

A. By Compactness

Compact Lie Groups

• $\mathrm{SO}(n)$, $\mathrm{SU}(n)$, $\mathrm{U}(n)$, $\mathrm{Sp}(n)$
• Spin groups $\mathrm{Spin}(n)$ — universal cover of $\mathrm{SO}(n)$
• Exceptional compact groups: $G_2$, $F_4$, $E_6$, $E_7$, $E_8$
• Tori $T^n = (S^1)^n = \mathrm{U}(1)^n$

Non-compact Lie Groups

• $\mathrm{GL}(n, \mathbb{R})$, $\mathrm{GL}(n, \mathbb{C})$
• $\mathrm{SL}(n, \mathbb{R})$, $\mathrm{SL}(n, \mathbb{C})$
• Indefinite orthogonal: $\mathrm{O}(p,q)$, $\mathrm{SO}(p,q)$, $\mathrm{SO}^+(p,q)$
• Indefinite unitary: $\mathrm{SU}(p,q)$, $\mathrm{Sp}(p,q)$
• Real/complex symplectic: $\mathrm{Sp}(2n, \mathbb{R})$, $\mathrm{Sp}(2n, \mathbb{C})$
• Lorentz group: $\mathrm{O}(3,1)$, $\mathrm{SO}^+(3,1)$
• de Sitter / anti-de Sitter groups

B. By Structure

Abelian Lie Groups

$\mathbb{R}^n, \quad T^n, \quad \mathbb{R}^k \times T^m, \quad \mathbb{C}^* \cong \mathbb{R}_{>0} \times S^1$

Nilpotent Lie Groups

• Heisenberg group: $H_{2n+1} = \left\{ \begin{pmatrix} 1 & \mathbf{x}^T & z \\ 0 & I_n & \mathbf{y} \\ 0 & 0 & 1 \end{pmatrix} : \mathbf{x}, \mathbf{y} \in \mathbb{R}^n, z \in \mathbb{R} \right\}$
• Unipotent matrix groups
• Carnot groups (stratified nilpotent)

Solvable Lie Groups

• Affine group $\mathrm{Aff}(\mathbb{R}^n) = \mathbb{R}^n \rtimes \mathrm{GL}(n, \mathbb{R})$
• Borel subgroups (maximal solvable)
• Upper triangular matrices

Semisimple Lie Groups

Simple Lie groups are classified by Dynkin diagrams:

Classical Series:

| Diagram | Group | Compact Form | |:-------:|:------|:-------------| | $A_n$ | $\mathrm{SL}(n+1)$ | $\mathrm{SU}(n+1)$ | | $B_n$ | $\mathrm{SO}(2n+1)$ | $\mathrm{SO}(2n+1)$ | | $C_n$ | $\mathrm{Sp}(2n)$ | $\mathrm{Sp}(n)$ | | $D_n$ | $\mathrm{SO}(2n)$ | $\mathrm{SO}(2n)$ |

Exceptional: $G_2, \quad F_4, \quad E_6, \quad E_7, \quad E_8$

Reductive Lie Groups

$G = (\text{semisimple}) \times (\text{central torus})$

Examples: $\mathrm{GL}(n)$, $\mathrm{U}(p,q)$

C. Classical Matrix Lie Groups — Reference Table

| Notation | Name | Field | Defining Condition | |:--------:|:-----|:-----:|:-------------------| | $\mathrm{GL}(n, \mathbb{F})$ | General linear | $\mathbb{R}, \mathbb{C}, \mathbb{H}$ | $\det A \neq 0$ | | $\mathrm{SL}(n, \mathbb{F})$ | Special linear | $\mathbb{R}, \mathbb{C}$ | $\det A = 1$ | | $\mathrm{O}(n)$ | Orthogonal | $\mathbb{R}$ | $A^T A = I$ | | $\mathrm{SO}(n)$ | Special orthogonal | $\mathbb{R}$ | $A^T A = I$, $\det A = 1$ | | $\mathrm{U}(n)$ | Unitary | $\mathbb{C}$ | $A^\dagger A = I$ | | $\mathrm{SU}(n)$ | Special unitary | $\mathbb{C}$ | $A^\dagger A = I$, $\det A = 1$ | | $\mathrm{Sp}(n)$ | Compact symplectic | $\mathbb{H}$ | $A^\dagger A = I$ (quaternionic) | | $\mathrm{Sp}(2n, \mathbb{R})$ | Real symplectic | $\mathbb{R}$ | $A^T J A = J$ | | $\mathrm{Sp}(2n, \mathbb{C})$ | Complex symplectic | $\mathbb{C}$ | $A^T J A = J$ | | $\mathrm{O}(p,q)$ | Indefinite orthogonal | $\mathbb{R}$ | $A^T \eta_{p,q} A = \eta_{p,q}$ | | $\mathrm{SU}(p,q)$ | Indefinite unitary | $\mathbb{C}$ | $A^\dagger \eta_{p,q} A = \eta_{p,q}$, $\det = 1$ | | $\mathrm{Spin}(n)$ | Spin | — | Double cover of $\mathrm{SO}(n)$ | | $\mathrm{Pin}(n)$ | Pin | — | Double cover of $\mathrm{O}(n)$ |

Where: $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}, \quad \eta_{p,q} = \mathrm{diag}(\underbrace{1,\ldots,1}_{p}, \underbrace{-1,\ldots,-1}_{q})$

D. Lie Groups in Physics

| Group | Role in Physics | |:------|:----------------| | $\mathrm{SO}(3)$ | Spatial rotations | | $\mathrm{SU}(2)$ | Spin, double cover of $\mathrm{SO}(3)$ | | $\mathrm{SO}(3,1)$ | Lorentz transformations | | $\mathrm{ISO}(3,1)$ | Poincaré group (spacetime symmetries) | | $\mathrm{ISO}(3)$ / $\mathrm{E}(3)$ | Euclidean group | | Galilean group | Non-relativistic spacetime | | $\mathrm{SO}(p+1, q+1)$ | Conformal group on $\mathbb{R}^{p,q}$ | | $\mathrm{U}(1)$ | Electromagnetism (QED) | | $\mathrm{SU}(2)$ | Weak force | | $\mathrm{SU}(3)$ | Strong force (QCD) | | $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ | Standard Model | | $\mathrm{SU}(5)$, $\mathrm{SO}(10)$, $E_6$, $E_8$ | Grand unified theories | | $\mathrm{Sp}(2n, \mathbb{R})$ | Classical phase space (Hamiltonian mechanics) | | $\mathrm{Diff}(M)$ | General covariance (GR) | | Loop groups $LG$ | Conformal field theory | | Virasoro group | 2D CFT, string theory | | Kac–Moody groups | Current algebras |

V. Algebraic Groups

A. Linear Algebraic Groups

Defined as Zariski-closed subgroups of $\mathrm{GL}(n, k)$:

• Unipotent groups — all eigenvalues equal to 1
• Diagonalizable / multiplicative groups
• Tori — split ($\mathbb{G}_m^n$), non-split, anisotropic
• Borel subgroups — maximal connected solvable
• Parabolic subgroups — contain a Borel subgroup
• Reductive algebraic groups — trivial unipotent radical
• Semisimple algebraic groups — trivial radical

B. Classification over Various Fields

| Base Field | Structure | |:-----------|:----------| | $\mathbb{C}$ | Matches complex Lie group classification | | $\mathbb{R}$ | Real forms of complex groups | | $\mathbb{Q}$ | Arithmetic groups, Galois cohomology | | $\mathbb{F}_q$ | Finite groups of Lie type | | $\mathbb{Q}_p$ | $p$-adic groups |

C. Abelian Varieties and Algebraic Tori

• Elliptic curves $(E, +)$ — genus 1 curves with group law
• Abelian varieties — higher-dimensional analogues
• Algebraic tori $T \cong (\mathbb{G}_m)^n$ (over algebraic closure)
• Néron models — extending abelian varieties over rings

VI. Infinite-Dimensional Groups

• Diffeomorphism groups $\mathrm{Diff}(M)$
• Symplectomorphism groups $\mathrm{Symp}(M, \omega)$
• Volume-preserving diffeomorphisms $\mathrm{Diff}_\mu(M)$
• Loop groups $LG = C^\infty(S^1, G)$
• Current algebras
• Kac–Moody groups:
  • Affine type $\tilde{\mathfrak{g}}$
  • Hyperbolic type
  • General type
• Gauge groups $\mathcal{G}(P)$ of principal bundles
• Unitary group of Hilbert space $\mathrm{U}(\mathcal{H})$
• Banach–Lie groups
• Fréchet–Lie groups

VII. Categorical and Generalized Structures

| Structure | Description | |:----------|:------------| | Groupoids | Categories with all morphisms invertible | | Lie groupoids | Smooth groupoids | | 2-groups | Categorical groups (monoidal groupoids) | | $\infty$-groupoids | Homotopy types | | Quantum groups | Hopf algebra deformations, e.g., $U_q(\mathfrak{g})$ | | Supergroups | $\mathbb{Z}_2$-graded, fermionic extensions | | Group schemes | Representable functors to groups | | Formal groups | Power series group laws |

VIII. Classification by Properties — Cross-Reference

| Property | Finite | Discrete Infinite | Lie | |:---------|:-------|:------------------|:----| | Abelian | $\mathbb{Z}/n$, products | $\mathbb{Z}^n$, torsion | $\mathbb{R}^n$, $T^n$ | | Simple | CFSG | Tarski monsters, etc. | Cartan classification | | Solvable | Burnside, Hall theory | Polycyclic | Triangular matrices | | Nilpotent | $p$-groups | f.g. nilpotent | Heisenberg | | Free | — | $F_n$ | — | | Perfect | $A_n$, simple groups | Many | Semisimple |

Appendix: Dynkin Diagrams

The simply-laced Dynkin diagrams (ADE):

A_n:    ○───○───○─ ⋯ ─○───○     (n nodes)

D_n:        ○
            │
        ○───○───○─ ⋯ ─○───○     (n nodes, n ≥ 4)

E_6:            ○
                │
        ○───○───○───○───○

E_7:            ○
                │
        ○───○───○───○───○───○

E_8:            ○
                │
        ○───○───○───○───○───○───○

Non-simply-laced diagrams:

B_n:    ○───○───○─ ⋯ ─○═══○     (n nodes, double bond at end)

C_n:    ○───○───○─ ⋯ ─○═══○     (n nodes, arrow reversed from B_n)

F_4:    ○───○═══○───○

G_2:    ○≡≡≡○                    (triple bond)

This taxonomy provides an extensive map of the "group zoo" across finite, infinite discrete, topological, and continuous settings.