groups

Groups

Abbreviation: Grp

Definition

A \emph{group} is a structure $\mathbf{G}=\langle G,\cdot ,^{-1},e\rangle $, where $\cdot $ 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

$\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

$\begin{array}{lr} f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &1
f(6)= &2
f(7)= &1
f(8)= &5
f(9)= &2
f(10)= &2
f(11)= &1
f(12)= &5
f(13)= &1
f(14)= &2
f(15)= &1
f(16)= &14
f(17)= &1
f(18)= &5
\end{array}$