−Table of Contents
Groups
Abbreviation: Grp
Definition
A \emph{group} is a structure G=⟨G,⋅,−1,e⟩, where ⋅ is an infix binary operation, called the \emph{group product}, −1 is a postfix unary operation, called the \emph{group inverse} and e is a constant (nullary operation), called the \emph{identity element}, such that
⋅ is associative: (xy)z=x(yz)
e is a left-identity for ⋅: ex=x
−1 gives a left-inverse: x−1x=e.
Remark: It follows that e is a right-identity and that −1gives a right inverse: xe=x, xx−1=e.
Morphisms
Let G and H be groups. A morphism from G to H is a function h:GarrowH that is a homomorphism:
h(xy)=h(x)h(y), h(x−1)=h(x)−1, h(e)=e
Examples
Example 1: ⟨SX,∘,−1,idX⟩, the collection of permutations of a sets X, with composition, inverse, and identity map.
Example 2: The general linear group ⟨GLn(V),⋅,−1,In⟩, the collection of invertible n×n matrices over a vector space V, with matrix multiplication, inverse, and identity matrix.
Basic results
Properties
Classtype | variety |
---|---|
Equational theory | decidable in polynomial time |
Quasiequational theory | undecidable |
First-order theory | undecidable |
Congruence distributive | no (Z2×Z2) |
Congruence modular | yes |
Congruence n-permutable | yes, n=2, p(x,y,z)=xy−1z is a Mal'cev term |
Congruence regular | yes |
Congruence uniform | yes |
Congruence types | 1=permutational |
Congruence extension property | no, consider a non-simple subgroup of a simple group |
Definable principal congruences | |
Equationally def. pr. cong. | no |
Amalgamation property | yes |
Strong amalgamation property | yes |
Epimorphisms are surjective | yes |
Locally finite | no |
Residual size | unbounded |
Finite members
f(1)=1f(2)=1f(3)=1f(4)=2f(5)=1f(6)=2f(7)=1f(8)=5f(9)=2f(10)=2f(11)=1f(12)=5f(13)=1f(14)=2f(15)=1f(16)=14f(17)=1f(18)=5
Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html