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#############################################################################
##
## coclass.gi Smallsemi - a GAP library of semigroups
## Copyright (C) 2008-2024 Andreas Distler & James D. Mitchell
##
## Licensing information can be found in the README file of this package.
##
#############################################################################
InstallGlobalFunction(NilpotentSemigroupsByCoclass,
function(arg...)
local n, d, r;
# INPUT: order <n>, coclass <d> (and rank <r>)
if Length(arg) < 2 or Length(arg) > 3 then
Error("number of arguments must be 2 or 3");
fi;
n := arg[1];
if not IsPosInt(n) then
Error("the order <n> must be a positive integer");
fi;
d := arg[2];
if not d in [0, 1, 2] then
Error("admissible values for the coclass <d> are 0, 1 and 2");
elif n <= d then
Info(InfoSmallsemi, 1, "Order does not exceed coclass.");
return [];
fi;
if Length(arg) = 3 then
r := arg[3];
if not r in [1, 2, 3] then
Error("admissible values for the rank <r> are 1, 2 and 3");
elif r = 1 and d <> 0 then
Info(InfoSmallsemi, 1, "Nilpotent semigroups are 1-generated ",
"if and only if they have coclass 0.");
return [];
elif d + 1 < r then
Info(InfoSmallsemi, 1,
"The rank <r> is larger than the coclass plus one, <d>+1.");
return [];
fi;
fi;
### cases covered by current implementation
if d = 0 then
return NilpotentSemigroupsCoclass0(n);
elif d = 1 then
return NilpotentSemigroupsCoclass1(n);
elif d = 2 then
if Length(arg) = 2 then
return NilpotentSemigroupsCoclass2(n);
elif Length(arg) = 3 then
if r = 2 then
return NilpotentSemigroupsCoclass2Rank2(n);
elif r = 3 then
return NilpotentSemigroupsCoclass2Rank3(n);
fi;
fi;
fi;
end);
## returns a list with the presentation for the semigroup of order <n> and
## coclass 0
InstallGlobalFunction(NilpotentSemigroupsCoclass0,
function(n)
local f, # free semigroup on one element
x; # generator of <f>
# catch input error
if not IsPosInt(n) then
Error("the size <n> has to be a positive integer");
fi;
f := FreeSemigroup(1);
x := GeneratorsOfSemigroup(f);
return [f / [[x[1] ^ n, x[1] ^ (n + 1)]]];
end);
## returns presentations for all semigroups of order <n> and coclass 1
## up to equivalence
InstallGlobalFunction(NilpotentSemigroupsCoclass1,
function(n)
local sgrps, # list of all semigroups
f, # free semigroup on two elements
x; # generators of <f>
# catch input error
if not IsPosInt(n) then
Error("the size <n> has to be a positive integer");
fi;
f := FreeSemigroup(2);
x := GeneratorsOfSemigroup(f);
# catch exceptional cases
if n <= 2 then
return [];
elif n = 3 then # the zero semigroup
return [f / [[x[1] ^ 2, x[1] ^ 3], # u ^ 2 = u ^ 3
[x[1] ^ 2, x[2] ^ 2], # u ^ 2 = v ^ 2
[x[1] ^ 2, x[1] * x[2]], # u ^ 2 = uv
[x[1] ^ 2, x[2] * x[1]]]]; # u ^ 2 = vu
fi;
sgrps := NilpotentSemigroupsCoclassD_NC(n, 1);
# additional semigroups since 3 - nilpotent
if n = 4 then
Add(sgrps, f / [[x[1] ^ 2, x[1] ^ 3], # u ^ 2 = u ^ 3
[x[1] ^ 2, x[2] ^ 2], # u ^ 2 = v ^ 2
[x[1] ^ 2, x[1] * x[2]], # u ^ 2 = uv
[x[1] ^ 2, x[2] ^ 3]]); # u ^ 2 = v ^ 3
Add(sgrps, f / [[x[1] ^ 2, x[1] ^ 3], # u ^ 2 = u ^ 3
[x[1] ^ 2, x[2] ^ 2], # u ^ 2 = v ^ 2
[x[1] * x[2], x[2] * x[1]], # uv = vu
[x[1] ^ 2, x[2] ^ 3]]); # u ^ 2 = v ^ 3
fi;
return sgrps;
end);
## returns presentations for all semigroups of order <n> and coclass 2
## up to equivalence
InstallGlobalFunction(NilpotentSemigroupsCoclass2,
function(n)
return Concatenation(NilpotentSemigroupsCoclass2Rank3(n),
NilpotentSemigroupsCoclass2Rank2(n));
end);
## returns presentations for all semigroups up to equivalence of order <n>
## and coclass <d> with <d>+1 generators and (at least?) <d> of those
## generating a monogenic semigroup of coclass <d>
InstallGlobalFunction(NilpotentSemigroupsCoclassD_NC,
function(n, d)
local sgrps, # list of all semigroups
f, # free semigroup on two elements
x, # generators of <f>
basrels, # relations appearing for each semigroup
rels, # relations of a specific semigroup
i, j, k, m; # loop counter
f := FreeSemigroup(d + 1);
x := GeneratorsOfSemigroup(f);
sgrps := EmptyPlist(5 * n + Int(n / 2) - 1); # HOW MANY?
basrels := [[x[1] ^ (n - d), x[1] ^ (n - d + 1)]];
for i in [2 .. d] do
Add(basrels, [x[1] ^ 2, x[i] ^ 2]); # u_1 ^ 2 = u_i ^ 2
Add(basrels, [x[1] ^ 2, x[1] * x[i]]); # u_1 ^ 2 = u_1u_i
Add(basrels, [x[1] ^ 2, x[i] * x[1]]); # u_1 ^ 2 = u_iu_1
for j in [i + 1 .. d] do
Add(basrels, [x[1] ^ 2, x[i] * x[j]]); # u_1 ^ 2 = u_iu_j
Add(basrels, [x[1] ^ 2, x[j] * x[i]]); # u_1 ^ 2 = u_ju_i
od;
od;
#
for k in [2 .. Int((n - d) / 2)] do
rels := ShallowCopy(basrels);
for i in [1 .. d] do
Add(rels, [x[i] * x[d + 1], x[d + 1] * x[i]]); # u_iv = vu_i
Add(rels, [x[i] * x[d + 1], x[1] ^ k]); # u_iv = u_1 ^ k
od;
Add(rels, [x[d + 1] ^ 2, x[1] ^ (2 * k - 2)]); # v ^ 2 = u_1 ^ (2k - 2)
Add(sgrps, f / rels);
od;
for k in [Int((n - d) / 2) + 1 .. n - d - 2] do
rels := ShallowCopy(basrels);
for i in [1 .. d] do
Add(rels, [x[i] * x[d + 1], x[d + 1] * x[i]]); # u_iv = vu_i
Add(rels, [x[i] * x[d + 1], x[1] ^ k]); # u_iv = u_1 ^ k
od;
Add(rels, [x[d + 1] ^ 2, x[1] ^ (n - d)]); # v ^ 2 = u_1 ^ (n - d)
Add(sgrps, f / rels);
Remove(rels);
# v ^ 2 = u_1 ^ (n - d - 1)
Add(rels, [x[d + 1] ^ 2, x[1] ^ (n - d - 1)]);
Add(sgrps, f / rels);
od;
# 0 <= j <= k <= m <= d
for j in [0 .. d] do
for k in [j .. d] do
for m in [Int((d + k + 1) / 2) .. d] do
rels := ShallowCopy(basrels);
for i in [1 .. j] do
Add(rels, [x[i] * x[d + 1], x[1] ^ (n - d - 1)]);
Add(rels, [x[d + 1] * x[i], x[1] ^ (n - d - 1)]);
od;
for i in [j + 1 .. k] do
Add(rels, [x[i] * x[d + 1], x[1] ^ (n - d)]);
Add(rels, [x[d + 1] * x[i], x[1] ^ (n - d)]);
od;
for i in [k + 1 .. m] do
Add(rels, [x[i] * x[d + 1], x[1] ^ (n - d)]);
Add(rels, [x[d + 1] * x[i], x[1] ^ (n - d - 1)]);
od;
for i in [m + 1 .. d] do
Add(rels, [x[i] * x[d + 1], x[1] ^ (n - d - 1)]);
Add(rels, [x[d + 1] * x[i], x[1] ^ (n - d)]);
od;
Add(rels, [x[d + 1] ^ 2, x[1] ^ (n - d)]); # v ^ 2 = u_1 ^ (n - d)
Add(sgrps, f / rels);
Remove(rels);
# v ^ 2 = u_1 ^ (n - d - 1)
Add(rels, [x[d + 1] ^ 2, x[1] ^ (n - d - 1)]);
Add(sgrps, f / rels);
od;
od;
od;
return sgrps;
end);
## returns presentations for all semigroups of order <n> and coclass <d>
## with <d>+1 generators and <d> of those generating a monogenic semigroup
## of coclass <d> up to equivalence
InstallGlobalFunction(NilpotentSemigroupsCoclassD,
function(n, d)
if not IsPosInt(n) then
Error("the size <n> has to be a positive integer");
elif not IsPosInt(d) then
Error("the coclass <d> has to be a positive integer");
# catch restriction of coclass value < d > depending on order < n >
elif n < 4 + d then
Error("the difference of the order <n> and the coclass <d> ",
" must be at least 4");
fi;
return NilpotentSemigroupsCoclassD_NC(n, d);
end);
## obtain f.p.-presentation for a nilpotent semigroup of degree 3 from
## the library
## AD: WORKS ONLY FOR 3-GENERATED SEMIGROUPS OF ORDER 5 AT THE MOMENT
InstallGlobalFunction(IsomorphicFpSemigroup,
function(sgrp)
local erz, # minimal generating set of <sgrp>
d, # minimal number of generators of <sgrp>
f, # free semigroup on <d> generators
x, # generating set of <f>
z, # multiplicative zero of <sgrp>
pairs, # list of 2-tuples with entries in [1 .. <d>]
pos, # position of <p> in unmodified <pairs>
p, # a pair that corresponds to a non - zero product in <sgrp>
rels, # relations of output semigroup isomorphic to < sgrp >
q; # loop counter, element in modified < pairs >
erz := MinimalGeneratingSet(sgrp);
d := Length(erz);
f := FreeSemigroup(d);
x := GeneratorsOfSemigroup(f);
pairs := Reversed(Tuples([1 .. d], 2));
z := MultiplicativeZero(sgrp);
pos := PositionProperty(pairs, p -> z <> erz[p[1]] * erz[p[2]]);
p := Remove(pairs, pos);
rels := EmptyPlist(d ^ 2);
Add(rels,
[x[p[1]] * x[p[1]] * x[p[2]], x[p[1]] * x[p[1]] * x[p[1]] * x[p[2]]]);
for q in pairs do
if z = erz[q[1]] * erz[q[2]] then
Add(rels, [x[p[1]] * x[p[1]] * x[p[2]], x[q[1]] * x[q[2]]]);
else
Add(rels, [x[p[1]] * x[p[2]], x[q[1]] * x[q[2]]]);
fi;
od;
return f / rels;
end);
## returns presentations for all semigroups of order <n> and coclass 2
## which have a minimal generating set of size 3 up to equivalence
##
## - implementing list from small coclass paper for n >= 6
## - using the semigroups from the library for n = 5
InstallGlobalFunction(NilpotentSemigroupsCoclass2Rank3,
function(n)
local sgrps, # list of all semigroups
f, # free semigroup on two elements
x, # generators of <f>
k, l, # loop counters
subsgrps, # list of candidates for V and W
v, w, # loop counters
V, W, # subsemigroups of size n-1 and coclass 1
Vgens, # generators of the subsemigroup V
Wgens, # generators of the subsemigroup W
Vrels, # relations of the subsemigroup V
Wrels, # relations of the subsemigroup W
rels; # relations of a semigroup
# catch input error
if not IsPosInt(n) then
Error("the size < n > has to be a positive integer");
fi;
f := FreeSemigroup(3);
x := GeneratorsOfSemigroup(f);
# catch exceptional cases
if n <= 3 then
return [];
elif n = 4 then
# the zero semigroup
return [f / [[x[1] ^ 2, x[1] ^ 3], # u ^ 2 = u ^ 3
[x[1] ^ 2, x[2] ^ 2], # u ^ 2 = v ^ 2
[x[1] ^ 2, x[1] * x[2]], # u ^ 2 = uv
[x[1] ^ 2, x[2] * x[1]], # u ^ 2 = vu
[x[1] ^ 2, x[3] ^ 2], # u ^ 2 = w ^ 2
[x[1] ^ 2, x[1] * x[3]], # u ^ 2 = uw
[x[1] ^ 2, x[3] * x[1]], # u ^ 2 = wu
[x[1] ^ 2, x[2] * x[3]], # u ^ 2 = vw
[x[1] ^ 2, x[3] * x[2]], # u ^ 2 = wv
[x[1] ^ 2, x[3] ^ 2]]]; # u ^ 2 = w ^ 2
elif n = 5 then
# just take the semigroups from the library
return List(AllSmallSemigroups(5,
Is3GeneratedSemigroup, true,
NilpotencyDegree, 3),
IsomorphicFpSemigroup);
fi;
if IsEvenInt(n) then
sgrps := EmptyPlist((21 * n ^ 2 + 22 * n - 96) / 8);
else
sgrps := EmptyPlist((21 * n ^ 2 + 36 * n - 81) / 8);
fi;
sgrps := NilpotentSemigroupsCoclassD_NC(n, 2);
# get semigroups of size n - 1 and coclass 1 .. .
subsgrps := NilpotentSemigroupsCoclass1(n - 1);
# .. . in which the two generators are not interchangeable
Remove(subsgrps, 1);
for v in [1 .. Length(subsgrps)] do
V := subsgrps[v];
# relations from V translated to new generators
Vgens := FreeGeneratorsOfFpSemigroup(V);
Vrels := List(RelationsOfFpSemigroup(V),
rel -> [MappedWord(rel[1], Vgens, x{[1, 2]}),
MappedWord(rel[2], Vgens, x{[1, 2]})]);
# u * v = u ^ k
k := ExtRepOfObj(Vrels[Length(Vrels) - 1][2])[2];
for w in [1 .. Length(subsgrps)] do
W := subsgrps[w];
# relations from W translated to new generators
Wgens := FreeGeneratorsOfFpSemigroup(W);
Wrels := List(RelationsOfFpSemigroup(W),
rel -> [MappedWord(rel[1], Wgens, x{[1, 3]}),
MappedWord(rel[2], Wgens, x{[1, 3]})]);
# first relation is already contained in <Vrels>
Remove(Wrels, 1);
# u * w = u ^ l
l := ExtRepOfObj(Wrels[Length(Wrels) - 1][2])[2];
rels := Concatenation(Vrels, Wrels);
# Case 2.d
if w >= v and IsCommutative(V) and IsCommutative(W) then
if k + l <= n - 2 then
# vw = wv
Add(rels, [x[2] * x[3], x[3] * x[2]]);
# vw = u ^ (k + l - 2)
Add(rels, [x[2] * x[3], x[1] ^ (k + l - 2)]);
Add(sgrps, f / rels);
else
# vw = u ^ (k + l - 2)
Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]);
# vw = u ^ (k + l - 2)
Add(rels, [x[2] * x[3], x[1] ^ (n - 3)]);
Add(sgrps, f / rels);
Remove(rels);
# vw = u ^ (k + l - 2)
Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]);
Add(sgrps, f / rels);
Remove(rels);
Remove(rels);
# vw = u ^ (k + l - 2)
Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]);
# vw = u ^ (k + l - 2)
Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]);
Add(sgrps, f / rels);
fi;
elif not IsCommutative(V) and IsCommutative(W) then
# se 2.b and part of Case 2.a
Add(rels, [x[2] * x[3], x[1] ^ (n - 3)]); # vw = u ^ (n - 3)
Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
Add(sgrps, f / rels);
Remove(rels);
Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
Add(sgrps, f / rels);
Remove(rels);
Remove(rels);
Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]); # vw = u ^ (n - 2)
Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
Add(sgrps, f / rels);
Remove(rels);
Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
Add(sgrps, f / rels);
elif v >= w and not IsCommutative(V) and not IsCommutative(W) then
# Case 2.a
Add(rels, [x[2] * x[3], x[1] ^ (n - 3)]); # vw = u ^ (n - 3)
Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
Add(sgrps, f / rels);
Remove(rels);
Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
Add(sgrps, f / rels);
Remove(rels);
Remove(rels);
Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]); # vw = u ^ (n - 2)
Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
Add(sgrps, f / rels);
Remove(rels);
Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
Add(sgrps, f / rels);
if v = w then
Remove(sgrps);
fi;
# take dual of W
Wrels[Length(Wrels) - 2][2] := x[1] ^ (n - 3);
Wrels[Length(Wrels) - 1][2] := x[1] ^ (n - 2);
rels := Concatenation(Vrels, Wrels);
Add(rels, [x[2] * x[3], x[1] ^ (n - 3)]); # vw = u ^ (n - 3)
Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
Add(sgrps, f / rels);
Remove(rels);
Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
Add(sgrps, f / rels);
Remove(rels);
Remove(rels);
Add(rels, [x[2] * x[3], x[1] ^ (n - 2)]); # vw = u ^ (n - 2)
Add(rels, [x[3] * x[2], x[1] ^ (n - 2)]); # wv = u ^ (n - 2)
Add(sgrps, f / rels);
Remove(rels);
Add(rels, [x[3] * x[2], x[1] ^ (n - 3)]); # wv = u ^ (n - 3)
Add(sgrps, f / rels);
fi;
od;
od;
return sgrps;
end);
## returns presentations for all semigroups of order <n> and coclass 2
## which have a minimal generating set of size 2 up to equivalence
##
## Implementing list from small coclass paper for n >= 7
InstallGlobalFunction(NilpotentSemigroupsCoclass2Rank2,
function(n)
local sgrps, # list of all semigroups
f, # free semigroup on two elements
x, # generators of <f>
rels, # relations of a semigroup
k; # loop counter
if not IsPosInt(n) then
Error("the size < n > has to be a positive integer");
fi;
# catch exceptional cases; for n > 4 see end of function
if n <= 4 then
return [];
fi;
f := FreeSemigroup(2);
x := GeneratorsOfSemigroup(f);
# might be 1 too big
sgrps := EmptyPlist(5 * n + Int(n / 2) - Int(n / 3) - 1);
# y = v^2 = uv = vu (third semigroup in Case 4, y = v^2)
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n - 2) = u ^ (n - 1)
[x[1] * x[2], x[2] * x[1]], # uv = vu
[x[2] ^ 2, x[1] * x[2]], # v ^ 2 = uv
[x[2] ^ 3, x[1] ^ (n - 3)]]); # v ^ 3 = u ^ (n - 3)
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n - 2) = u ^ (n - 1)
[x[1] * x[2], x[2] * x[1]], # uv = vu
[x[2] ^ 2, x[1] * x[2]], # v ^ 2 = uv
[x[2] ^ 3, x[1] ^ (n - 2)]]); # v ^ 3 = u ^ (n - 2)
# y = uv = vu
for k in [3, 4 .. Int((n - 1) / 2)] do
# u ^ (n - 2) = u ^ (n - 1)
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)],
[x[1] * x[2], x[2] * x[1]], # uv = vu
[x[2] ^ 2, x[1] ^ (2 * k - 4)], # v ^ 2 = u ^ (2k - 4)
[x[1] ^ 2 * x[2], x[1] ^ k]]); # u ^ 2v = u ^ k
od;
for k in [n - Int(n / 2) .. n - 2] do
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[2] * x[1]], # uv=vu
[x[2] ^ 2, x[1] ^ (n - 4)], # v^2 =u^(n-4)
[x[1] ^ 2 * x[2], x[1] ^ k]]); # u^2v=u^k
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[2] * x[1]], # uv=vu
[x[2] ^ 2, x[1] ^ (n - 3)], # v^2=u^(n-3)
[x[1] ^ 2 * x[2], x[1] ^ k]]); # u^2v=u^k
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n - 1)
[x[1] * x[2], x[2] * x[1]], # uv=vu
[x[2] ^ 2, x[1] ^ (n - 2)], # v^2=u^(n-2)
[x[1] ^ 2 * x[2], x[1] ^ k]]); # u^2v=u^k
od;
# y = uv = v ^ 2
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[2] ^ 2, x[1] * x[2]], # v^2=uv
[x[2] * x[1], x[1] ^ 2], # vu=u^2
[x[1] * x[2] ^ 2, x[1] ^ 3]]); # uv^2=u^3
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[2] ^ 2, x[1] * x[2]], # v^2=uv
[x[2] * x[1], x[1] ^ (n - 3)], # vu=u^(n-3)
[x[1] * x[2] ^ 2, x[1] ^ (n - 2)]]); # uv^2=u^(n-2)
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[2] ^ 2, x[1] * x[2]], # v^2=uv
[x[2] * x[1], x[1] ^ (n - 2)], # vu=u^(n-2)
[x[1] * x[2] ^ 2, x[1] ^ (n - 2)]]); # uv^2=u^(n-2)
# y = v ^ 2
for k in [2 .. n - 2] do
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[2] * x[1]], # uv=vu
[x[1] * x[2], x[1] ^ k], # uv=u^k
[x[2] ^ 3, x[1] ^ (3 * k - 3)]]); # v^3=u^(3k-3)
od;
for k in [n - Int(2 * n / 3) .. n - 2] do
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[2] * x[1]], # uv=vu
[x[1] * x[2], x[1] ^ k], # uv=u^k
[x[2] ^ 3, x[1] ^ (n - 3)]]); # v^3=u^(n-3)
od;
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[1] ^ (n - 3)], # uv=u^(n-3)
[x[2] * x[1], x[1] ^ (n - 2)], # vu=u^(n-2)
[x[2] ^ 3, x[1] ^ (n - 3)]]); # v^3=u^(n-3)
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[1] ^ (n - 3)], # uv=u^(n-3)
[x[2] * x[1], x[1] ^ (n - 2)], # vu=u^(n-2)
[x[2] ^ 3, x[1] ^ (n - 2)]]); # v^3=u^(n-2)
# y=vu
for k in [2 .. n - Int((n + 1) / 2) - 1] do
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[1] ^ k], # uv=u^k
[x[2] ^ 2, x[1] ^ (2 * k - 2)], # v^2=u^(2k-2)
[x[2] * x[1] ^ 2, x[1] ^ (k + 1)]]); # vu^2=u^(k+1)
od;
for k in [n - Int((n + 1) / 2) .. n - 2] do
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[1] ^ k], # uv=u^k
[x[2] ^ 2, x[1] ^ (n - 2)], # v^2=u^(n-2)
[x[2] * x[1] ^ 2, x[1] ^ (k + 1)]]); # vu^2=u^(k+1)
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[1] ^ k], # uv=u^k
[x[2] ^ 2, x[1] ^ (n - 3)], # v^2=u^(n-3)
[x[2] * x[1] ^ 2, x[1] ^ (k + 1)]]); # vu^2=u^(k+1)
od;
for k in [n - 3 .. n - 2] do
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[1] ^ (n - 4)], # uv=u^(n-4)
[x[2] ^ 2, x[1] ^ k], # v^2=u^k
[x[2] * x[1] ^ 2, x[1] ^ (n - 2)]]); # vu^2=u^(n-2)
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[1] ^ (n - 3)], # uv=u^(n-3)
[x[2] ^ 2, x[1] ^ k], # v^2=u^k
[x[2] * x[1] ^ 2, x[1] ^ (n - 3)]]); # vu^2=u^(n-3)
Add(sgrps, f / [[x[1] ^ (n - 1), x[1] ^ (n - 2)], # u^(n-2)=u^(n-1)
[x[1] * x[2], x[1] ^ (n - 2)], # uv=u^(n-2)
[x[2] ^ 2, x[1] ^ k], # v^2=u^k
[x[2] * x[1] ^ 2, x[1] ^ (n - 3)]]); # vu^2=u^(n-3)
od;
if n = 5 then
sgrps := sgrps{[17, 15, 5, 18, 16, 4, 10, 8, 2, 6]};
Add(sgrps, f / [[x[1] ^ 2, x[1] ^ 2 * x[2]], # u^2=u^2v
[x[2] ^ 2, x[1] ^ 2 * x[2]], # v^2=u^2v
[x[1] * x[2] * x[1], x[1] ^ 2 * x[2]], # uvu=u^2v
[x[2] * x[1] * x[2], (x[1] * x[2]) ^ 2]]); # vuv=(uv)^2
fi;
if n = 6 then
sgrps := sgrps{[23, 21, 8, 30, 29, 5, 6, 20, 3, 24, 22, 7, 27, 26, 14,
11, 19, 2, 10, 13, 4, 17, 18, 16, 1, 9]};
rels := [[x[1] ^ 2, x[1] ^ 3], # u^2=u^3
[x[1] ^ 2, x[2] ^ 2], # u^2=v^2
[x[1] ^ 2, x[1] * x[2] * x[1]], # u^2=uvu
[x[1] * x[2] * x[1], x[2] * x[1] * x[2]], # uvu=vuv
[x[1] ^ 2, x[2] ^ 3], # u^2=v^3
[x[2] ^ 2, x[2] * x[1] * x[2]], # v^2=vuv
[x[1] ^ 2, x[2] * x[1]], # u^2=vu
[x[2] * x[1], x[1] * x[2] ^ 2], # vu=uv^2
[x[2] ^ 2, x[1] * x[2] * x[1]], # v^2=uvu
[x[1] ^ 2, x[2] * x[1] * x[2]], # u^2=vuv
[x[2] ^ 2, x[2] * x[1]], # v^2=vu
[x[1] * x[2], x[2] * x[1]], # uv=vu
[x[1] ^ 2 * x[2], x[1] ^ 2], # u^2v=u^2
[x[1] ^ 3, x[1] * x[2] * x[1]], # u^3=uvu
[x[1] ^ 3, x[2] ^ 3], # u^3=v^3
[x[1] * x[2] ^ 2, x[1] ^ 2], # uv^2=u^2
[x[2] * x[1], x[2] * x[1] ^ 2], # vu=vu^2
[x[1] ^ 2 * x[2], x[1] ^ 3], # u^2v=u^3
[x[1] * x[2], x[1] ^ 2], # uv=u^2
[x[2] * x[1] ^ 3, x[1] ^ 3]]; # vu^3=u^3
Add(sgrps, f / rels{[1, 2, 3]}); # [6, 42]
Add(sgrps, f / rels{[1, 2, 4]}); # [6, 141]
Add(sgrps, f / rels{[3, 5, 6]}); # [6, 338]
Add(sgrps, f / rels{[1, 5, 7]}); # [6, 413]
Add(sgrps, f / rels{[1, 5, 8]}); # [6, 414]
Add(sgrps, f / rels{[5, 9, 10]}); # [6, 481]
Add(sgrps, f / rels{[1, 5, 11]}); # [6, 583]
Add(sgrps, f / rels{[5, 6, 9]}); # [6, 659]
Add(sgrps, f / rels{[1, 5, 12, 13]}); # [6, 880]
Add(sgrps, f / rels{[2, 10, 14]}); # [6, 1485]
Add(sgrps, f / rels{[2, 3, 10]}); # [6, 1865]
Add(sgrps, f / rels{[15, 16, 17]}); # [6, 2416]
Add(sgrps, f / rels{[7, 15, 16]}); # [6, 2417]
Add(sgrps, f / rels{[11, 15, 16]}); # [6, 2494]
Add(sgrps, f / rels{[12, 15, 16, 18]}); # [6, 2556]
Add(sgrps, f / rels{[11, 19, 20]}); # [6, 2558]
Add(sgrps, f / rels{[11, 12, 20]}); # [6, 2559]
fi;
return sgrps;
end);
[ 0.49Quellennavigators
Projekt
]
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