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## 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: $xe=x$, $xx^{-1}=e$.

This definition shows that groups form a variety.

##### 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.

### Properties

Classtype variety decidable in polynomial time undecidable undecidable no ($\mathbb{Z}_{2}\times \mathbb{Z}_{2}$) yes yes, n=2, $p(x,y,z)=xy^{-1}z$ is a Mal'cev term yes yes 1=permutational no, consider a non-simple subgroup of a simple group no yes yes yes no 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