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.

(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>

Computations of Braid groups

(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

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}, bab^{-1} = 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>

Free groups

F(n) = <a_{1},...,a_{n} | a_{i}a_{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_{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

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