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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: x1x=e.

Remark: It follows that e is a right-identity and that 1gives a right inverse: xe=x, xx1=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(x1)=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)=xy1z 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

Subclasses

Superclasses

References


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