# Difference between revisions of "Group Examples"

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==== <div id="vonDyck_groups">von Dyck groups</div> ==== | ==== <div id="vonDyck_groups">von Dyck groups</div> ==== | ||

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

− | ==== <div id="Free_abelian_group">Free abelian | + | ==== <div id="Free_abelian_group">Free abelian groups</div> ==== |

− | Z | + | Z(n) = <a_1,...,a_n | [a_i,a_j] = 1 for all i,j> |

− | ==== <div id=" | + | ==== <div id="Free_groups">Free groups</div> ==== |

F(n) = <a_1,...,a_n | a_ia_i^{-1} = a_i^{-1}a_i = 1> | F(n) = <a_1,...,a_n | a_ia_i^{-1} = a_i^{-1}a_i = 1> | ||

==== <div id="Fibonacci_groups">Fibonacci groups</div> ==== | ==== <div id="Fibonacci_groups">Fibonacci groups</div> ==== | ||

− | + | Fibonacci groups are related to the inductive definition of the Fibonacci numbers f_{i} + f_{i+1} = f_{i+2} where f_{1} = f_{2} = 1. These groups have the following finite presentation: | |

F(2,m) = <x_1,...,x_m | x_{i}x_{i+1} = x_{i+2}> | F(2,m) = <x_1,...,x_m | x_{i}x_{i+1} = x_{i+2}> | ||

− | ==== <div id=" | + | ==== <div id="Heisenberg_groups">Heisenberg groups</div> ==== |

− | 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> | + | 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> |

==== <div id="Higman_group">Higman group</div> ==== | ==== <div id="Higman_group">Higman group</div> ==== | ||

− | 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> | + | 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> |

− | ==== <div id=" | + | ==== <div id="Ordinary_tetrahedon_groups">Ordinary tetrahedon groups</div> ==== |

− | The | + | The Ordinary tetrahedon 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> | 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> | ||

− | ==== <div id=" | + | ==== <div id="Thompson_group">Thompson group</div> ==== |

− | T = <a,b | [ab^{-1},a^{-1}ba] = [ab^{-1},a^{-2}ba^2] = 1> | + | 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_nx_k = x_{n+1} for all k < n> with a = x_0, x_n = a^{1-n}ba^{n-1} | |

− | |||

==== <div id="Triangle_groups">Triangle groups</div> ==== | ==== <div id="Triangle_groups">Triangle groups</div> ==== | ||

− | The | + | 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} | 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: | There are three different cases depending on the choice of the parameters l,m,n: |

## Revision as of 15:50, 1 July 2013

### Non-abelian Groups

#### Baumslag groups

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

BS(m,n) = <a,b | b*a^m = a^n*b>

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

(Reference: G. Baumslag and D. Solitar, Some two generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. , 689 (1962) pp. 199–201.)

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>

#### 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>

(Reference: W. Magnus, Braid groups: A survey, Proceedings of the Second International Conference on the Theory of Groups, Canberra, Australia, 1973, pp. 463-487.)

#### Cyclic groups

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

#### Dicyclic groups

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

#### Dihedral groups

The dihedral group of degree n is the group of symmetries of a regular polynom. This non-abelian group consists of 2n elements, n rotations and n reflections. Let r be a single rotation and s be an arbitrary reflection. Then the group has the following representation

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

#### 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>

#### Free abelian groups

Z(n) = <a_1,...,a_n | [a_i,a_j] = 1 for all i,j>

#### Free groups

F(n) = <a_1,...,a_n | a_ia_i^{-1} = a_i^{-1}a_i = 1>

#### Fibonacci groups

Fibonacci groups are related to the inductive definition of the Fibonacci numbers f_{i} + f_{i+1} = f_{i+2} where f_{1} = f_{2} = 1. These groups have the following finite presentation:

F(2,m) = <x_1,...,x_m | x_{i}x_{i+1} = x_{i+2}>

#### 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>

#### 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>

#### Ordinary tetrahedon groups

The Ordinary tetrahedon 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>

#### 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_nx_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