Group Examples

Non-abelian Groups

Checklist of groups

Baumslag groups

Baumslag-Solitar groups are examples of two-generator one-relator groups.

BS(m,n) = <a,b | ba^{m} = a^{n}b> 

where m, n are natural numbers. Computations of Baumslag groups.

Another variation of the Baumslag groups, called the Baumslag-Gersten group, is defined by:

BG = <a,b | (b^{-1}a^{-1}b)a(b^{-1}a^{-1}b) = a^{2}>

Computations of Baumslag-Gersten groups.

Braid groups

For a natural number n, the following is a presentation of the Braid group with n-1 generators and n strands.

B(n) = <g_{1},...,g_{n-1} | g_{i}g_{j} = g_{j}g_{i} for |i-j| >= 2, g_{i}g_{i+1}g_{i} = g_{i+1}g_{i}g_{i+1} for 1 <= i <= n-2>

Computations of Braid groups

Artin groups (Generalized Braid groups)

An Artin group (Generalized Braid group) is a group with a representation of the form

G = <x_{1},...,x_{n} | <x_1,x_2>^{m_{1,2}} = <x_2,x_1>^{m_{2,1}}, ... , <x_{n-1},x_{n}>^{m_{n-1,n}} = <x_{n},x_{n-1}>^{m_{n,n-1}}>

where m_{i,j} = m_{j,i} > 1

If m is not infinity <x_i,x_j>^{m} denotes an alterning product of x_i and x_j of length m beginning with x_i.

For example <x_1,x_2>^{4} = x_1x_2_x1_x2

If m is infinity there is no relation between x_i and x_j.

Computations of Artin groups

Cyclic groups

For a natural number n > 1 the cyclic groups can be represented as

C(n) = <a | a^{n} = 1>

Computations of cyclic groups

Dicyclic groups

Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}>

Computations of dicyclic groups

Dihedral groups

Dih(n) = <r,s | r^{n} = s^{2} = s^{-1}rs = r^{-1}>

Computations of dihedral groups

von Dyck groups

The von Dyck groups are sometimes referred to as ordinary triangle groups and are subgroups of index 2 in Triangle(l, m, n) generated by words of even length in the generators a, b, c. A specific representation is given for x = ab, y = ca, yx = cb:

D(l,m,n) = <x,y | x^{l} = y^{m} = (xy)^{n} = 1>

Computations of von Dyck groups

Free abelian groups

Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j>

Computations of free abelian groups

Free groups

F(n) = <a_{1},...,a_{n} | a_{i}a_{i}^{-1} = a_{i}^{-1}a_{i} = 1>

Computations of free groups

Fibonacci groups

These groups have the following finite presentation:

F(2,m) = <x_{1},...,x_{m} | x_{i}x_{i+1} = x_{i+2}>

where the indices are taken modulo m. Computations of Fibonacci groups

Heisenberg groups

H(2k+1) = <a_{1},...,a_{k},b_{1},...,b_{k},c | [a_{i},b_{i}] = c, [a_{i},c] = [b_{i},c], [a_{i},b_{j}] = 1 for all i != j>

Computations of Heisenberg groups

Higman group

H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}>

Computations of Higman groups

Ordinary tetrahedron groups

The Ordinary tetrahedron groups are groups with the following presentation where e_i >= 2 and f_i >= 2 for all i.

G(e_1,e_2,e_3,f_1,f_2,f_3) = <x,y,z | x^{e_1} = y^{e_2} = z^{e_3} = (xy^{-1})^{f_1} = (yz^{-1})^{f_2} = (zx^{-1})^{f_3} = 1>

Computations of Ordinary Tetrahedron groups

Lamplighter group

The Lamplighter group has the following presentation

 G = <a,b | (ab^{n}ab^{-n})^2 = 1>

for all n in Z. Computations of Lamplighter group

Symmetric groups

The symmetric groups have the following presentation for n > 0

 S_n = <a_{1},..,a_{n-1} | a_{i}^2 = 1, a_{i}a_{j} = a_{j}a_{i} for j != i +/- 1, (a_{i}a_{i+1})^3 = 1>

where a_{i} is the neighbor transposition a_{i} = (i,i+1) Computations of symmetric groups

Tetraeder group

The Tetraeder group has the following presentation

 A_4 = <a,b | a^2 = b^3 = (ab)^3 = 1>

Computations of the Tetraeder group

Oktaeder group

The Oktaeder group has the following presentation

 O = <a,b | a^2 = b^3 = (ab)^4 = 1>

Computations of the Oktaeder group

Ikosaeder group

The Ikosaeder group has the following presentation

 I = <a,b | a^2 = b^3 = (ab)^5 = 1>

Computations of the Ikosaeder group

Quaternion group

The Quaternion group has the following presentation

 Q_8 = <a,b | a^4 = 1, x^2 = y^2, y^{-1}xy = x^{-1}>

Computations of the Quaternion group

Tits group

The Tits group has the following presentation

 T = <a,b | a^2 = b^3 = (ab)^13 = [a,b]^5 = [a,bab]^4 = ((ab)^4ab^{-1})^6 = 1 >

Computations of the Tits group

Special Linear group

The Tits group has the following presentation

 SL_2(Z) = <a,b | aba = bab, (aba)^4 = 1 >

Computations of the Special Linear group

Modular group

The Tits group has the following presentation

 PSL(2,Z) = <a,b | a^2 = (ab)^3 = 1 >

Computations of the Modular group

Thompson group

T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^{2}] = 1>
  = <x_{0},x_{1},x_{2},... | x_{k}^{-1}x_{n}x_{k} = x_{n+1} for all k < n> with a = x_{0}, x_{n} = a^{1-n}ba^{n-1}

Triangle groups

The triangle groups describe the application of reflections across the sides of a triangle (A,B,C) with the three reflections a,b,c and can be represented as the following:

Triangle(l,m,n) = {a,b,c | a^{2} = b^{2} = c^{2} = (ab)^{l} = (bc)^{m} = (ca)^{n} = 1}

There are three different cases depending on the choice of the parameters l,m,n:

The euclidian case:    1/l + 1/m + 1/n = 1
The spherical case:    1/l + 1/m + 1/n > 1
The hyperbolical case: 1/l + 1/m + 1/n < 1

Old Data at Ftp

ftp://apcocoa.org/pub/symbolic_data