groups - MathStructures

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=====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====

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

Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html

====Subclasses====
[[p-groups]] 

[[nilpotent groups]] 

[[solvable groups]] 

====Superclasses====
[[Monoids]] 

[[Inverse semigroups]] 


====References====

[(Ln19xx>
)]
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