List of Groups
Introduction
This article lists the groups the corpus introduces, from the cyclic group $\mathbb{Z}/n\mathbb{Z}$ to the Monster, together with the data that separates them: the order, the centre and the derived subgroup where the corpus records them. The groups fall into the families of the object ladder — cyclic, abelian, dihedral, symmetric and alternating — into the matrix and general linear groups, into the free and presented groups of combinatorial group theory, into the solvable and nilpotent groups, into the finite simple groups of the classification, and into the groups that arithmetic, geometry and $K$-theory attach to their objects.
Every entry points to the article that introduces the group and names the invariant recorded there. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being a group or near-group that fails one property its family has, with the failure named and the article that records it.
Cyclic, Abelian and Dihedral Groups
The lowest groups are the cyclic ones, finite and infinite, and their direct sums; the abelian groups are classified by the structure theorem in the finitely generated case and by the divisible and reduced invariants in general. The dihedral group is the first non-abelian group in the corpus, generated by a rotation and a reflection.
| Group | The property it has | Introduced in |
|---|---|---|
| $(\mathbb{Z},+)$ | the infinite cyclic group, abelian, with trivial derived subgroup | Groups, §1 |
| $\mathbb{Z}/n\mathbb{Z}$ | the cyclic group of order $n$, abelian | Groups, §1 |
| $C_n = \langle x \mid x^n = 1\rangle$ | the cyclic group of order $n$, $Z(C_n) = C_n$, $[C_n,C_n] = 1$ | Finite Groups and Symmetry |
| Finite abelian groups | direct sums of cyclic groups, as the structure theorem classifies them | Finitely Generated Abelian Groups |
| Divisible abelian groups | abelian groups in which every element is divisible, classified by their torsion | Infinite Abelian Groups |
| Free abelian group $\mathbb{Z}^{(I)}$ | the abelian group with a basis, with rank $|I|$ | Infinite Abelian Groups |
| $\mathbb{Z}_p$ | the additive group of $p$-adic integers, profinite and torsion-free | Profinite Groups and the Krull Topology |
| $D_n$ | the dihedral group of order $2n$; $Z(D_n)$ is trivial for odd $n$ and of order $2$ for even $n$; $[D_n,D_n] = \langle r^2\rangle$ | Finite Groups and Symmetry |
| $Q_8$ | the quaternion group of order $8$; $Z(Q_8) = \{\pm1\}$, $[Q_8,Q_8] = \{\pm1\}$, every subgroup normal | Groups, §20 |
| Group of order $p^2$ | abelian, isomorphic to $\mathbb{Z}/p^2\mathbb{Z}$ or $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ | Groups, §13 |
| Group of order $pq$, $p < q$ | cyclic unless $p \mid q-1$, in which case a non-abelian semidirect product occurs | Groups, §13 |
| Non-example: $(\mathbb{Z}\setminus\{0\}, \cdot)$ | a monoid that fails inverses: only $\pm1$ are invertible | Groups, §1 |
Symmetric and Alternating Groups
The symmetric group $S_n$ is the group of bijections of an $n$-element set, of order $n!$, and the alternating group $A_n$ is its subgroup of even permutations, of order $n!/2$ and normal in $S_n$. For $n \geq 3$ the symmetric group is non-abelian with trivial centre and derived subgroup $A_n$, and for $n \geq 5$ the alternating group is simple; these two facts supply the first two families of the classification.
| Group | The property it has | Introduced in |
|---|---|---|
| $S_n$ | the symmetric group of order $n!$; $Z(S_n) = 1$ and $[S_n,S_n] = A_n$ for $n \geq 3$ | Groups, §1 |
| $S_n^{\mathrm{ab}}$ | the abelianisation, isomorphic to $\mathbb{Z}/2\mathbb{Z}$ for $n \geq 3$ | Groups, §9 |
| $A_n$ | the alternating group of order $n!/2$, normal in $S_n$ | Groups |
| $A_n$, $n \geq 5$ | a finite simple group, the alternating family of the classification | The Classification of Finite Simple Groups |
| $S_2$, $S_3$ | $S_2 \cong C_2$; $S_3 \cong D_3$ | Finite Groups and Symmetry |
| Non-example: $S_n$, $n \geq 3$ | a group that fails commutativity | Groups, §1 |
| Non-example: $A_4$ | a group of order $12$ that fails simplicity: its Klein four-subgroup is normal | Finite Groups and Symmetry |
Matrix and Classical Groups
The general linear group $GL_n(F)$ is the group of invertible $n \times n$ matrices over a field, or invariantly the automorphism group of a vector space; its centre is the group of scalar matrices and its derived subgroup is the special linear group, with named exceptions. The special linear group $SL_n(F)$ is the kernel of the determinant, and $PGL_n$ and $PSL_n$ are the quotients by the scalars. The classical groups that preserve a form are deferred to Part II.
| Group | The property it has | Introduced in |
|---|---|---|
| $GL_n(F)$ | the group of invertible matrices, the automorphism group of $F^n$; centre the scalar matrices | The General Linear Group |
| $SL_n(F)$ | the kernel of the determinant; the commutator subgroup of $GL_n(F)$ with named exceptions | The Special Linear Group and the Determinant |
| $PGL_n(F)$ | the quotient of $GL_n(F)$ by its scalar matrices | The General Linear Group |
| $PSL_n(F)$ | the quotient of $SL_n(F)$ by its scalar matrices, simple for the large cases | Finite Simple Groups of Lie Type |
| $\operatorname{Aff}(V) = V \rtimes GL(V)$ | the affine group, the translations extended by the linear group | Affine Spaces and Translations |
| $R^\times$ | the group of units of a ring | Groups, §1 |
| $M_n(k)^\times$ | the units of a matrix algebra, equal to $GL_n(k)$ | Matrix Algebras |
| Warning: $O(V,q)$, $U(V,h)$, $Sp(V,\omega)$, $\operatorname{Spin}$, $\operatorname{Pin}$ | not group-layer objects: each is defined by a form and a distance, so it belongs to Part II | The General Linear Group; Finite Groups and Symmetry |
Free, Presented and Infinite Groups
A free group $F(X)$ is the universal group on a set, equivalently the group presented by the generators $X$ with no relations; its abelianisation is the free abelian group, and its centre is trivial when the rank is at least two. A presentation $\langle X \mid R\rangle$ presents the quotient of the free group by the normal closure of the relations, the free product is the coproduct of groups, and the amalgamated product and the HNN extension are its relatives with a shared subgroup.
| Group | The property it has | Introduced in |
|---|---|---|
| Free group $F(X)$ | the universal group on a set; $F(X)^{\mathrm{ab}} = \mathbb{Z}^{(X)}$ | Generators, Presentations and Free Products |
| $F_2$ | the free group of rank $2$; $[F_2,F_2] = \ker(F_2 \to \mathbb{Z}^2)$, and $Z(F_2) = 1$ | Generators, Presentations and Free Products |
| Presented group $\langle X \mid R\rangle$ | the quotient of $F(X)$ by the normal closure of $R$ | Generators, Presentations and Free Products |
| Free product $G_1 * G_2$ | the coproduct of groups | Generators, Presentations and Free Products |
| $D_\infty = C_2 * C_2$ | the infinite dihedral group | Generators, Presentations and Free Products |
| Free product with amalgamation $G_1 *_H G_2$ | the pushout of two groups over a common subgroup | Combinatorial Group Theory |
| HNN extension | the group presented by a stable letter conjugating one subgroup onto another | Combinatorial Group Theory |
| $PSL_2(\mathbb{Z})$ | the modular group, $C_2 * C_3$, with $SL_2(\mathbb{Z}) \cong C_4 *_{C_2} C_6$ | Generators, Presentations and Free Products |
| $BS(2,3)$ | the Baumslag–Solitar group $\langle a,t \mid ta^2t^{-1} = a^3\rangle$, the standard non-Hopfian example | Generators, Presentations and Free Products |
| Braid group $B_n$ | the group of $n$ strands, presented by the Artin generators, surjecting onto $S_n$ | Braid Groups |
| Pure braid group $P_n$ | the kernel of $B_n \to S_n$ | Braid Groups |
| Coxeter group | the group presented by reflections with the braid and involution relations | Coxeter Groups |
| Finite Coxeter groups | the crystallographic and non-crystallographic finite types, classified by their diagrams | Coxeter Groups |
| Thompson groups $F, T, V$ | the groups of PL homeomorphisms of the interval, circle and Cantor set | Thompson Groups and the Cantor Set |
| Warning: Weyl and reflection groups | require a form and a distance, so they belong to Part II | Coxeter Groups |
| Non-example: the free abelian group of rank $\geq 2$ | not a free group: it satisfies a relation $[a,b]=1$ | Generators, Presentations and Free Products |
Solvable, Nilpotent and Finite Simple Groups
A solvable group has a finite derived series reaching the identity and a nilpotent group a finite lower central series; the $p$-groups are nilpotent, and the Hall subgroups are the analogues of the Sylow subgroups in the solvable case. The finite simple groups are the building blocks of all finite groups, and the classification names them: the cyclic groups of prime order, the alternating groups of degree at least five, the groups of Lie type and the twenty-six sporadic groups.
| Group | The property it has | Introduced in |
|---|---|---|
| Solvable group | a group whose derived series terminates at $1$ | Solvable and Nilpotent Groups |
| Nilpotent group | a group whose lower central series terminates at $1$ | Solvable and Nilpotent Groups |
| $p$-group | a group of order $p^k$, nilpotent, with nontrivial centre | Groups, §13 |
| Hall subgroup | a subgroup whose order and index are coprime, the solvable analogue of a Sylow subgroup | Solvable and Nilpotent Groups |
| Fitting and Frattini subgroups | the largest normal nilpotent subgroup; the intersection of the maximal subgroups | Solvable and Nilpotent Groups |
| Finite simple group | a group with no nontrivial proper normal subgroup | The Classification of Finite Simple Groups |
| Cyclic group of prime order | the first family of the classification | The Classification of Finite Simple Groups |
| Group of Lie type | one of the sixteen families of Chevalley, Steinberg and Suzuki–Ree groups | Finite Simple Groups of Lie Type |
| Sporadic simple group | one of the twenty-six groups outside the three infinite families | The Classification of Finite Simple Groups |
| Monster $\mathbb{M}$ | the largest sporadic simple group, containing twenty of the sporadics as subquotients | The Classification of Finite Simple Groups |
| Baby Monster $\mathbb{B}$ | the second largest sporadic group, the centraliser of an involution in $\mathbb{M}$ | The Classification of Finite Simple Groups |
| Non-example: $S_5$ | a group that fails solvability: it contains the simple $A_5$ | Solvable and Nilpotent Groups |
| Non-example: $BS(2,3)$ | a group that fails Hopfianity: it is isomorphic to a proper quotient of itself | Generators, Presentations and Free Products |
Groups of Arithmetic, Geometry and K-Theory
Several groups are attached to objects rather than presented: the ideal class group of a Dedekind domain, the Brauer group of a field, the group law of an elliptic curve, the Grothendieck groups of a ring, and the Galois groups of a field extension. The last is the automorphism group of the extension and is profinite in the infinite case, which is the subject of the article on the Krull topology.
| Group | The property it has | Introduced in |
|---|---|---|
| Ideal class group $\operatorname{Cl}(K)$ | the finite abelian group of fractional ideals modulo principal ideals | Dedekind Domains and Ideal Class Groups |
| Brauer group $\operatorname{Br}(F)$ | the abelian group of central simple algebras over $F$ up to Morita equivalence | Central Simple Algebras and the Brauer Group |
| $E(K)$ | the group law on an elliptic curve; abelian, finitely generated over a number field by Mordell–Weil | Elliptic Curves |
| $K_0(R)$, $K_1(R)$ | the Grothendieck group of projective modules and the determinant group | K-Theory of Rings |
| $\mathcal{O}_K^\times$ | the unit group of a ring of integers, finitely generated abelian by Dirichlet's theorem | Algebraic Number Theory |
| Galois group $\operatorname{Gal}(L/K)$ | the automorphisms of $L$ fixing $K$; of order $[L:K]$ when the extension is finite and Galois | Galois Theory |
| Absolute Galois group $G_K$ | the Galois group of a separable closure, profinite | Profinite Groups and the Krull Topology |
| Profinite group | an inverse limit of finite groups, compact and totally disconnected | Profinite Groups and the Krull Topology |
| $\hat{\mathbb{Z}}$ | the profinite completion of $\mathbb{Z}$, $\prod_p \mathbb{Z}_p$ | Profinite Groups and the Krull Topology |
| Non-example: $\operatorname{Gal}(L/K)$ for a non-Galois extension | the automorphism group fails to have order $[L:K]$: the extension is not normal or not separable | Galois Theory |
Summary
The list gathers the groups of the corpus. The cyclic, abelian and dihedral groups include $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, the finite and infinite abelian groups, $\mathbb{Z}_p$, $D_n$ and $Q_8$, with the groups of order $p^2$ and $pq$. The symmetric and alternating groups are $S_n$ and $A_n$, the latter simple for $n \geq 5$. The matrix and classical groups are $GL_n$, $SL_n$, $PGL_n$, $PSL_n$, $\operatorname{Aff}(V)$ and the unit groups, and the classical groups that preserve a form are deferred to Part II. The free, presented and infinite groups are the free groups, the presented groups, the free, amalgamated and HNN constructions, the modular group, the Baumslag–Solitar group, the braid groups, the Coxeter groups and the Thompson groups. The solvable and nilpotent groups lead to the finite simple groups — cyclic of prime order, alternating, of Lie type and sporadic, up to the Monster. The arithmetic, geometric and $K$-theoretic groups are the ideal class group, the Brauer group, the group law of an elliptic curve, the $K$-groups, the unit group of a ring of integers, the Galois groups and the profinite groups. Each non-example — $(\mathbb{Z}\setminus\{0\},\cdot)$, $S_n$ for $n\geq3$, $A_4$, the free abelian group of rank at least two, $S_5$, $BS(2,3)$ and the Galois group of a non-Galois extension — names the property that fails.
Summary of Notation
The article denotes its objects by name; the symbols appearing in the tables are the standard ones of the introducing articles.
| Symbol | Meaning |
|---|---|
| $C_n$, $D_n$, $S_n$, $A_n$, $Q_8$ | cyclic, dihedral, symmetric, alternating and quaternion groups |
| $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{Z}_p$, $\hat{\mathbb{Z}}$ | integers, cyclic group, $p$-adic integers, profinite integers |
| $F(X)$, $F_2$, $G_1*G_2$, $G_1*_H G_2$ | free group, rank-two free group, free and amalgamated products |
| $\langle X \mid R\rangle$ | presentation |
| $B_n$, $P_n$ | braid and pure braid groups |
| $GL_n$, $SL_n$, $PGL_n$, $PSL_n$ | general and special linear groups, projective versions |
| $\operatorname{Aff}(V)$ | affine group $V \rtimes GL(V)$ |
| $R^\times$ | group of units |
| $\operatorname{Cl}(K)$, $\operatorname{Br}(F)$, $E(K)$, $K_0$, $K_1$ | class group, Brauer group, elliptic-curve group, K-groups |
| $\operatorname{Gal}(L/K)$, $G_K$ | Galois group and absolute Galois group |
| $Z(G)$, $[G,G]$, $G^{\mathrm{ab}}$ | centre, derived subgroup, abelianisation |
| $\mathbb{M}$, $\mathbb{B}$ | Monster and Baby Monster |
Further Reading
- Joseph J. Rotman, An Introduction to the Theory of Groups (Springer, 4th ed. 1995), for the cyclic, symmetric, alternating, dihedral and matrix groups and their centres and derived subgroups.
- Derek J. S. Robinson, A Course in the Theory of Groups (Springer, 2nd ed. 1996), for the solvable, nilpotent and infinite groups and the structure theory.
- Daniel Gorenstein, Richard Lyons and Ronald Solomon, The Classification of the Finite Simple Groups (American Mathematical Society, 1994–), for the four families of the classification and the sporadic groups.
- John Conway, Robert Curtis, Simon Norton, Richard Parker and Robert Wilson, Atlas of Finite Groups (Oxford University Press, 1985), for the orders, centres and maximal subgroups of the finite simple groups.
- Roger Lyndon and Paul Schupp, Combinatorial Group Theory (Springer, 1977), for the free, presented, amalgamated and HNN constructions.