Group Examples

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

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