# Difference between revisions of "Group Examples"

Andraschko (talk | contribs) (extended description) |
|||

(167 intermediate revisions by 6 users not shown) | |||

Line 1: | Line 1: | ||

− | + | 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. | |

− | |||

− | |||

− | |||

− | |||

− | + | [[:ApCoCoA:GroupsToCheck|Checklist of Groups]] | |

+ | |||

+ | [[:ApCoCoA:XML-Codes|Examples in Symbolic Data Format]] | ||

+ | ==== <div id="Baumslag_groups">Baumslag Groups</div> ==== | ||

+ | 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. [[:ApCoCoA:Baumslag groups|Computations of Baumslag Groups]]. | ||

Another variation of the Baumslag groups, called the Baumslag-Gersten group, is defined by: | 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> | + | BG = <a,b | (b^{-1}a^{-1}b)a(b^{-1}a^{-1}b) = a^{2}> |

+ | [[:ApCoCoA:Baumslag-Gersten groups|Computations of Baumslag-Gersten Groups]]. | ||

+ | |||

+ | ==== <div id="Braid_groups">Braid Groups</div> ==== | ||

+ | 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> | ||

+ | [[:ApCoCoA:Braid groups|Computations of Braid Groups]] | ||

+ | |||

+ | ==== <div id="Artin_groups">Artin Groups (Generalized Braid Groups)</div> ==== | ||

+ | 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. | ||

+ | |||

+ | [[:ApCoCoA:Artin groups|Computations of Artin Groups]] | ||

+ | |||

+ | ==== <div id="Cyclic_groups">Cyclic Groups</div> ==== | ||

+ | For a natural number n > 1 the cyclic groups can be represented as | ||

+ | C(n) = <a | a^{n} = 1> | ||

+ | [[:ApCoCoA:Cyclic groups|Computations of Cyclic Groups]] | ||

+ | |||

+ | ==== <div id="Dicyclic_groups">Dicyclic Groups</div> ==== | ||

+ | Dic(n) = <a,b | a^{2n} = 1, a^{n} = b^{2}, b^{-1}ab = a^{-1}> | ||

+ | |||

+ | [[:ApCoCoA:Dicyclic groups|Computations of Dicyclic Groups]] | ||

+ | |||

+ | ==== <div id="Dihedral_groups">Dihedral Groups</div> ==== | ||

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

+ | [[:ApCoCoA:Dihedral groups|Computations of Dihedral Groups]] | ||

− | ==== | + | ==== <div id="vonDyck_groups">von Dyck Groups</div> ==== |

− | + | 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> | |

− | + | [[:ApCoCoA:VonDyck groups|Computations of von Dyck Groups]] | |

+ | ==== <div id="Free_abelian_group">Free Abelian Groups</div> ==== | ||

+ | Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j> | ||

+ | [[:ApCoCoA:FreeAbelian groups|Computations of Free Abelian Groups]] | ||

− | ==== <div id=" | + | ==== <div id="Free_groups">Free Groups</div> ==== |

− | + | F(n) = <a_{1},...,a_{n} | a_{i}a_{i}^{-1} = a_{i}^{-1}a_{i} = 1> | |

+ | [[:ApCoCoA:Free groups|Computations of Free Groups]] | ||

+ | ==== <div id="Fibonacci_groups">Fibonacci Groups</div> ==== | ||

+ | 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. | ||

+ | [[:ApCoCoA:Fibonacci groups|Computations of Fibonacci Groups]] | ||

− | ==== <div id=" | + | ==== <div id="Hecke_groups">Hecke Groups</div> ==== |

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

+ | [[:ApCoCoA:Hecke groups|Computations of Hecke groups]] | ||

+ | ==== <div id="Extended Hecke_groups">Extended Hecke Groups</div> ==== | ||

+ | 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> | ||

+ | [[:ApCoCoA:Extended Hecke groups|Computations of Extended Hecke Groups]] | ||

− | ==== <div id=" | + | ==== <div id="Heisenberg_groups">Heisenberg Groups</div> ==== |

− | + | 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> | |

− | + | [[:ApCoCoA:Heisenberg groups|Computations of Heisenberg Groups]] | |

+ | ==== <div id="Higman_group">Higman Group</div> ==== | ||

+ | 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}> | ||

+ | [[:ApCoCoA:Higman groups|Computations of Higman Group]] | ||

− | ==== <div id=" | + | ==== <div id="Ordinary_tetrahedron_groups">Ordinary Tetrahedron Groups</div> ==== |

− | The | + | 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> | |

+ | [[:ApCoCoA:OrdinaryTetrahedron groups|Computations of Ordinary Tetrahedron Groups]] | ||

+ | ==== <div id="Lamplighter_group">Lamplighter Group</div> ==== | ||

+ | The Lamplighter group has the following presentation | ||

+ | G = <a,b | (ab^{n}ab^{-n})^2 = 1> | ||

+ | for all n in Z. | ||

+ | [[:ApCoCoA:Lamplighter group|Computations of Lamplighter Group]] | ||

− | ==== <div id=" | + | ==== <div id="Symmetric_groups">Symmetric Groups</div> ==== |

− | + | 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) | ||

+ | [[:ApCoCoA:Symmetric groups|Computations of Symmetric Groups]] | ||

+ | ==== <div id="Tetraeder_group">Tetraeder Group</div> ==== | ||

+ | The Tetraeder group has the following presentation | ||

+ | A_4 = <a,b | a^2 = b^3 = (ab)^3 = 1> | ||

+ | [[:ApCoCoA:Tetraeder group|Computations of the Tetraeder Group]] | ||

− | ==== <div id=" | + | ==== <div id="Oktaeder_group">Oktaeder Group</div> ==== |

− | + | The Oktaeder group has the following presentation | |

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

+ | [[:ApCoCoA:Oktaeder group|Computations of the Oktaeder Group]] | ||

+ | ==== <div id="Ikosaeder_group">Ikosaeder Group</div> ==== | ||

+ | The Ikosaeder group has the following presentation | ||

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

+ | [[:ApCoCoA:Ikosaeder group|Computations of the Ikosaeder Group]] | ||

− | ==== <div id=" | + | ==== <div id="Mathieu11_group">Mathieu Group M_{11}</div> ==== |

− | The | + | 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> | |

+ | [[:ApCoCoA:Mathieu11 group|Computations of the Mathieu Group M_{11}]] | ||

+ | ==== <div id="Mathieu22_group">Mathieu Group M_{22}</div> ==== | ||

+ | 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> | ||

+ | [[:ApCoCoA:Mathieu22 group|Computations of the Mathieu Group M_{22}]] | ||

− | ==== <div id=" | + | ==== <div id="Quaternion_group">Quaternion Group</div> ==== |

− | + | The Quaternion group has the following presentation | |

+ | Q_8 = <a,b | a^4 = 1, x^2 = y^2, y^{-1}xy = x^{-1}> | ||

+ | [[:ApCoCoA:Quaternion group|Computations of the Quaternion Group]] | ||

+ | ==== <div id="Tits_group">Tits Group</div> ==== | ||

+ | 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 > | ||

+ | [[:ApCoCoA:Tits group|Computations of the Tits Group]] | ||

− | ==== <div id=" | + | ==== <div id="SpecialLinear_group">Special Linear Group</div> ==== |

− | + | The special linear group has the following presentation | |

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

+ | [[:ApCoCoA:SpecialLinear group|Computations of the Special Linear Group]] | ||

+ | ==== <div id="SpecialLinear32_group">Special Linear Group SL_2(32)</div> ==== | ||

+ | 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 > | ||

+ | [[:ApCoCoA:SpecialLinear32 group|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. | ||

− | ==== <div id=" | + | ==== <div id="SpecialLinearPrime_group">Special Linear Group with Prime p</div> ==== |

− | The | + | 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> | |

+ | [[:ApCoCoA:SpecialLinearPrime group|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 | ||

+ | ==== <div id="Modular_group">Modular Group</div> ==== | ||

+ | The Modular group has the following presentation | ||

+ | PSL(2,Z) = <a,b | a^2 = (ab)^3 = 1 > | ||

+ | [[:ApCoCoA:Modular group|Computations of the Modular Group]] | ||

− | ==== <div id=" | + | ==== <div id="Alternating_groups">Alternating Groups</div> ==== |

− | + | 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> | |

− | + | [[:ApCoCoA:Alternating groups|Computations of the Alternating Groups]] | |

+ | ==== <div id="Thompson_group">Thompson Group</div> ==== | ||

+ | 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} | ||

+ | [[:ApCoCoA:Thompson group|Computation of the Thompson Group]] | ||

− | ==== <div id="Triangle_groups">Triangle | + | ==== <div id="Triangle_groups">Triangle Groups</div> ==== |

− | The | + | 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} | + | 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: | 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 euclidian case: 1/l + 1/m + 1/n = 1 | ||

Line 76: | Line 175: | ||

The hyperbolical case: 1/l + 1/m + 1/n < 1 | The hyperbolical case: 1/l + 1/m + 1/n < 1 | ||

+ | [[:ApCoCoA:Triangle groups|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> | ||

+ | [[:ApCoCoA:Picard group|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> | ||

+ | [[:ApCoCoA:Euclidean Bianchi groups|Computations of Euclidean Bianchi Groups]]. | ||

+ | |||

+ | ==== <div id="Knot_groups">Knot Groups</div> ==== | ||

+ | 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 > | ||

+ | |||

+ | [[:ApCoCoA:Knot Group|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 | ||

+ | [[:ApCoCoA: Torus Knot Group|Computation of the torus knot group]] | ||

+ | |||

+ | ==== <div id="Coxeter_groups">Coxeter Groups</div> ==== | ||

+ | 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 > | ||

+ | [[:ApCoCoA:Coxeter groups|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> | ||

+ | [[:ApCoCoA:Coxeter groupsH|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> | ||

+ | [[:ApCoCoA:Coxeter Group F4|Computation of the Coxeter Group F4]] | ||

+ | |||

+ | ==== <div id="Other_groups">Other Groups</div> ==== | ||

+ | 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. | ||

+ | [[:ApCoCoA:Other1 groups|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> | ||

+ | [[:ApCoCoA:Other2 groups|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> | ||

+ | [[:ApCoCoA:Other4 groups|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> | ||

+ | [[:ApCoCoA:Other5 groups|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. | ||

+ | |||

+ | [[:ApCoCoA:Other7 groups|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}> | ||

+ | [[:ApCoCoA:Other8 groups|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}> | ||

+ | [[:ApCoCoA:Other9 groups|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 | ||

+ | |||

+ | [[:ApCoCoA:Other11 groups|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. | ||

+ | [[:ApCoCoA:Other12 groups|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> | |

+ | [[:ApCoCoA:Other13 groups|Computations of the other Group 13]] |

## Latest revision as of 15:49, 29 October 2020

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.

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>

#### 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>

#### 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}>

#### 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>

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>