Group Examples

The following are examples of finitely presented groups. For each of them, we offer the ApCoCoA code to compute the Gröbner basis of the defining ideal of their group ring.

Checklist of Groups

Examples in Symbolic Data Format

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} = 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

Hecke Groups

 H(lambda_q) = <x,y | x^2=(xy)^q=1, for q >= 3>

Computations of Hecke groups

Extended Hecke Groups

The Extended Hecke Group is a kind of Hecke group with the following presentation:

<R,X,Y | R^2 = X^2 = Y^p = (RX)^2 = (YR)^2 = 1>

Computations of Extended Hecke 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 Group

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_{11} has the following presentation

 M_{11} = <a,b | a^2 = b^4 = (ab)^11 = (ab^2)^6 = ababab^{−1}abab^2ab^{−1}abab^{−1}ab^{−1} = 1>

Computations of the Mathieu Group M_{11}

Mathieu Group M_{22}

The Mathieu group M_{22} has the following presentation

 M_{22} = <a,b | a^2 = b^4 = (ab)^11 = (ab^2)^5 = [a,bab]^3 = (ababab^{−1})^5 = 1>

Computations of the Mathieu Group M_{22}

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}

Computation of the Thompson Group

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

Euclidean Bianchi Groups

There are five cases of Euclidean Bianchi groups: Eb_{1}, Eb_{2}, Eb_{3}, Eb_{7}, Eb_{11}.

The group Eb_{1} is called Picard Group:

 Eb_{1} = <a,l,t,u | tu = ut, a^{2} = l^{2} = (al)^{2} = (tl)^{2} = (ul)^{2} = (at)^{3} = (ual)^3 = 1>

Computation of Picard Group.

Others are specifiable by their indices:

 Eb_{3} = <a,d,e | a^{2} = (da)^{2} = (ade)^{2} = d^{3} = e^{3} = (dae^{-1})^{3} = e^{-1}dedaed^{-1}e^{-1}d^{-1}a = 1>

 Eb_{2} = <a,t,u | tu = ut, a^{2} = (at)^{3} = (u^{-1}aua)^{2} = 1>
 Eb_{7} = <a,t,u | tu = ut, a^{2} = (at)^{3} = (u^{-1}auat)^{2} = 1>
 Eb_{11} = <a,t,u | tu = ut, a^{2} = (at)^{3} = (u^{-1}auat)^{3} = 1>

Computations of Euclidean Bianchi Groups.

Knot Groups

There are a lot of cases of Knot groups.

One of them is the "figure eight group" which has the following presentation:

feg(x,y)= < x,y | yxy^{-1}xy = xyx^{-1}yx >

Computation of the figure eight group

Another is the "torus knot group" which has the following presantation:

tng(a,b)= < a,b| a^p = b^q = 1 >
where p,q are relatively prime

Computation of the torus knot group

Coxeter Groups

There different types of Coxeter groups:

One of them, is the H3 Coxter Group, called the full icosahedral group, with the following presentation:

 H3 = <x,y,z | x^2 = y^2 = z^2 = (xy)^2 = (xz)^2 =(yz)^2 = 1 >

Computation of the full Icosahedral Group

Another Coxeter Group is the group H4 with the presentation:

H4 = <v,x,y,z | v^2 = x^2 = y^2 = z^2 = (vx)^3 =(vy)^2 = (vz)^2 =(xy)^4 = (xz)^2 =(yz)^3 = 1>

Computation of the Coxeter Group H4

Third Coxeter Group is the group F4 with the presentation:

F4 = <v,x,y,z | v^2 = x^2 = y^2 = z^2 = (vx)^3 =(vy)^2 = (vz)^2 =(xy)^4 = (xz)^2 =(yz)^3 = 1>

Computation of the Coxeter Group F4

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

The next group consists of two generators and two relations (#generators = #relations) with the following representation:

11) G = <x,t | xt^{r} = tx^{r},t^{n} = 1>

for r >= 1 and n >= 2

Computations of the other Group 11

Group number 12 has the following finite representation:

12) G = <x,t | tx^{a}t^{-1} = x^{b},t^{n} = 1>

for a,b >= 1 and n >= 2. In this case (like group number 11 above) the count of generators is equal to the count of relations.

Computations of the other Group 12

For the next group we need four parameters: a,b,c,d. It is represented by this representation:

13) G = <x,y | x^2 = xy^{a}xy^{b}xy^{c}xy^{d} = 1>

Computations of the other Group 13