Group Examples
Non-abelian 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>
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.
Cyclic groups
For a natural number n > 1 the cyclic groups can be represented as
C(n) = <a | a^{n} = 1>
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} = 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>
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
Hecke groups
H(lambda_q) = <x,y | x^2=(xy)^q=1, for q >= 3>
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}>
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
Mathieu group M_{11}
The Mathieu group M_{22} has the following presentation
M_{11} = <x,y | xy = y^{-3}x^{-4}, y[x^{-1},y^{-1}] = [x^{-1},y]x^{-1}>
Computations of the Mathieu group M_{11} References: EFFICIENT PRESENTATIONS FOR THREE SIMPLE GROUPS P.E. Kenne Department of Computer Science, Australian National University, GPO Box 4, Canberra ACT 2601.
Mathieu group M_{22}
The Mathieu group M_{22} has the following presentation
M_{22} = <a,b | a^2 = (ab)^{11}, (ababb)^7 = b^4, (ab)^{2}(ab^{-1})^{2}abb(ab)^{2}ab^{-1}ab(abb)^2=b>
Computations of the Mathieu group M_{22} References: Efficient presentations for the Mathieu simple group M 22 and its cover Marston Conder, George Havas and Colin Ramsay
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 special linear group has the following presentation
SL_2(Z) = <a,b | aba = bab, (aba)^4 = 1 >
Computations of the Special Linear group
Special Linear group SL_2(32)
The special linear group S_2(32) has the following presentation
SL_2(32) = <a,b | b^3 = (ab)^{2} = a^{6}ba^{-2}ba^{2}b^{-1}a^{3}b^{-1}ab^{-1}a^{-3}b >
Computations of the Special Linear group SL_2(32) Reference: EFFICIENT PRESENTATIONS FOR THREE SIMPLE GROUPS P.E. Kenne Department of Computer Science, Australian National University, GPO Box 4, Canberra ACT 2601.
Special Linear group with prime p
The Special Linear group SL_2(p) has the following presentation
SL_2(p) = (x,y | x^{2} =(xy)^{3},(xy^{4}xy^{t})^{2}y^{p}x^{2k}=1>
Computations of the Special Linear group with prime p Reference: EFFICIENT PRESENTATIONS FOR FINITE SIMPLE GROUPS AND RELATED GROUPS Colin M. Campbell, E.F. Robertson and P.D. Williams* 1988
Modular group
The Modular group has the following presentation
PSL(2,Z) = <a,b | a^2 = (ab)^3 = 1 >
Computations of the Modular group
Alternating groups
The Alternating groups have the following presentation
A_{n+2} = <x_{1},..x_{n} | x_{i}^{3} = (x_{i}x_{j})^2 = 1 for every i != j>
Computations of the Alternating groups
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
Computations of the Triangle groups
Other groups
The following groups are either special cases of the groups above or hasn't been named yet.
The first example is a group which has the same count of generators and relations (#Generators = #Relations) and is solvable with length 6.
1) G = <a,b | a^{2}b^{-6} = (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1>
where k is congruent to 3 mod 6.
Computations of the other group 1
The next group, denoted by G, is the largest finite generalized triangle group and called the Rosenberger Monster. This group has an order |G| = 2^{20}*3^{4}*5.
2) G = <a,b | a^2 = b^3 = (abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1>
Levai, Rosenberger and Souvignier showed that G is finite an the group below, denoted by H, is infinite.
3) H = <a,b | a^2 = b^3 = (abababab^{2}abab^{2}ab^{2})^2 = 1>
Computations of the other groups 2 and 3
This group is an example of a generalized triangle group with order |G| = 1440
4) G = <a,b | a^2 = b^3 = (ababab^{2})^{3} = 1>
Computations of the other group 4
The following groups, denoted by G and H, are both solvable with length 4 but differs in the factors. G has an order |G| = 4224.
5) G = <a,b | a^{2}b^{-4} = (ababab^{3})^{2} = 1>
6) H = <a,b | a^{2}b^{4} = (ababab^{3})^{2} = 1>
Computations of the other groups 5 and 6
The next group denoted by G has the following representation:
7) G = <a,b | a^{2}b^{-3} = (ababa^{2}ab^{2})^2 = 1>
It holds that |G| = 9216 and that G is solvable of length 4.
Computations of the other group 7
The next group, denoted by H(r,n,s) has more relations and depends on three parameters. The finite representation for r > s and s >= 0 is given by:
8) H(r,n,s) = <a_{1},a_{2},...,a_{n} | a_{1}a_{2}..a_{r} = a_{r+1}a_{r+2}..a_{r+s}, a_{2}a_{3}..a_{r+1} =
a_{r+2}a_{r+3}..a_{r+s+1},..,a_{n}a_{1}..a_{r-1} = a_{r}a_{r+1}..a_{r+s-1}>
Computations of the other group 8
This group has the following finite representation:
9) F(r,n,k) = <a_{1},a_{2},..,a_{n} | a_{1}a_{2}..a_{r} = a_{r+k},a_{2}a_{3}..a_{r+1} = a_{r+k+1},..,a_{n}a_{1}a_{2}..a_{r-1} =
a_{r+k-1}>
It follow that the group number 8 (denoted by H(r,n,s)) is isomorphic to the group F(r,n,k) for s = k = 1.
A special case which is also isomorphic to H(r,n,s) is the group denoted by F(r,n). A finite representation is given by:
10) F(r,n) = <a_{1},..,a_{n} | a_{1}a_{2}..a_{r} = a_{r+1},..,a_{n}a_{1}..a_{r-1} = a_{r}>
Computations of the other groups 9 and 10
