Groups
Abbreviation: Grp
Definition
A group is a structure $\mathbf{G}=\langle G,\cdot ,^{-1},e\rangle $, where $\cdot $ is an infix binary operation, called the group product, $^{-1}$ is a postfix unary operation, called the group inverse and $e$ is a constant (nullary operation), called the identity element, such that
$\cdot $ is associative: $(xy)z=x(yz)$
$e$ is a left-identity for $\cdot$: $ex=x$
$^{-1}$ gives a left-inverse: $x^{-1}x=e$.
Remark: It follows that $e$ is a right-identity and that $^{-1}$gives a right inverse: $xe=x$, $xx^{-1}=e$.
Morphisms
Let $\mathbf{G}$ and $\mathbf{H}$ be groups. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:Garrow H$ that is a homomorphism:
$h(xy)=h(x)h(y)$, $h(x^{-1})=h(x)^{-1}$, $h(e)=e$
Examples
Example 1: $\langle S_{X},\circ ,^{-1},id_{X}\rangle $, the collection of permutations of a sets $X$, with composition, inverse, and identity map.
Example 2: The general linear group $\langle GL_{n}(V),\cdot ,^{-1},I_{n}\rangle $, the collection of invertible $n\times n$ 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 ($\mathbb{Z}_{2}\times \mathbb{Z}_{2}$) |
Congruence modular | yes |
Congruence n-permutable | yes, n=2, $p(x,y,z)=xy^{-1}z$ 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
$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &2\\ f(7)= &1\\ f(8)= &5\\ f(9)= &2\\ f(10)= &2\\ f(11)= &1\\ f(12)= &5\\ f(13)= &1\\ f(14)= &2\\ f(15)= &1\\ f(16)= &14\\ f(17)= &1\\ f(18)= &5\\ \end{array}$
Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html
Subclasses
Superclasses
References
Trace: » groups