Difference between revisions of "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}>

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}>

Dihedral Groups

Dih(n) = <r,s | r^{n} = s^{2} = s^{-1}rs = r^{-1} = 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_{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>

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>

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>

Oktaeder Group

The Oktaeder group has the following presentation

O = <a,b | a^2 = b^3 = (ab)^4 = 1>

Ikosaeder Group

The Ikosaeder group has the following presentation

I = <a,b | a^2 = b^3 = (ab)^5 = 1>

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>

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>

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}>

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 >

Special Linear Group

The special linear group has the following presentation

SL_2(Z) = <a,b | aba = bab, (aba)^4 = 1 >

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 >

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>

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

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>

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>

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.

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>

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>

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>

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.

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}>

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}>

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

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.

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>