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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra permgrps.spad}
\author{Gerhard Schneider, Holger Gollan, Johannes Grabmeier, M. Weller}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain PERMGRP PermutationGroup}
<<domain PERMGRP PermutationGroup>>=
)abbrev domain PERMGRP PermutationGroup
++ Authors: G. Schneider, H. Gollan, J. Grabmeier
++ Date Created: 13 February 1987
++ Date Last Updated: 24 May 1991
++ Basic Operations:
++ Related Constructors: PermutationGroupExamples, Permutation
++ Also See: RepresentationTheoryPackage1
++ AMS Classifications:
++ Keywords: permutation, permutation group, group operation, word problem
++ References:
++ C. Sims: Determining the conjugacy classes of a permutation group,
++ in Computers in Algebra and Number Theory, SIAM-AMS Proc., Vol. 4,
++ Amer. Math. Soc., Providence, R. I., 1971, pp. 191-195
++ Description:
++ PermutationGroup implements permutation groups acting
++ on a set S, i.e. all subgroups of the symmetric group of S,
++ represented as a list of permutations (generators). Note that
++ therefore the objects are not members of the \Language category
++ \spadtype{Group}.
++ Using the idea of base and strong generators by Sims,
++ basic routines and algorithms
++ are implemented so that the word problem for
++ permutation groups can be solved.
--++ Note: we plan to implement lattice operations on the subgroup
--++ lattice in a later release
PermutationGroup(S:SetCategory): public == private where
L ==> List
PERM ==> Permutation
FSET ==> Set
I ==> Integer
NNI ==> NonNegativeInteger
V ==> Vector
B ==> Boolean
OUT ==> OutputForm
SYM ==> Symbol
REC ==> Record ( orb : L NNI , svc : V I )
REC2 ==> Record(order:NNI,sgset:L V NNI,_
gpbase:L NNI,orbs:L REC,mp:L S,wd:L L NNI)
REC3 ==> Record(elt:V NNI,lst:L NNI)
REC4 ==> Record(bool:B,lst:L NNI)
public ==> SetCategory with
coerce : % -> L PERM S
++ coerce(gp) returns the generators of the group {\em gp}.
generators : % -> L PERM S
++ generators(gp) returns the generators of the group {\em gp}.
elt : (%,NNI) -> PERM S
++ elt(gp,i) returns the i-th generator of the group {\em gp}.
random : (%,I) -> PERM S
++ random(gp,i) returns a random product of maximal i generators
++ of the group {\em gp}.
random : % -> PERM S
++ random(gp) returns a random product of maximal 20 generators
++ of the group {\em gp}.
++ Note: {\em random(gp)=random(gp,20)}.
order : % -> NNI
++ order(gp) returns the order of the group {\em gp}.
degree : % -> NNI
++ degree(gp) returns the number of points moved by all permutations
++ of the group {\em gp}.
base : % -> L S
++ base(gp) returns a base for the group {\em gp}.
strongGenerators : % -> L PERM S
++ strongGenerators(gp) returns strong generators for
++ the group {\em gp}.
wordsForStrongGenerators : % -> L L NNI
++ wordsForStrongGenerators(gp) returns the words for the strong
++ generators of the group {\em gp} in the original generators of
++ {\em gp}, represented by their indices in the list, given by
++ {\em generators}.
coerce : L PERM S -> %
++ coerce(ls) coerces a list of permutations {\em ls} to the group
++ generated by this list.
permutationGroup : L PERM S -> %
++ permutationGroup(ls) coerces a list of permutations {\em ls} to
++ the group generated by this list.
orbit : (%,S) -> FSET S
++ orbit(gp,el) returns the orbit of the element {\em el} under the
++ group {\em gp}, i.e. the set of all points gained by applying
++ each group element to {\em el}.
orbits : % -> FSET FSET S
++ orbits(gp) returns the orbits of the group {\em gp}, i.e.
++ it partitions the (finite) of all moved points.
orbit : (%,FSET S)-> FSET FSET S
++ orbit(gp,els) returns the orbit of the unordered
++ set {\em els} under the group {\em gp}.
orbit : (%,L S) -> FSET L S
++ orbit(gp,ls) returns the orbit of the ordered
++ list {\em ls} under the group {\em gp}.
++ Note: return type is L L S temporarily because FSET L S has an error.
-- (GILT DAS NOCH?)
member? : (PERM S, %)-> B
++ member?(pp,gp) answers the question, whether the
++ permutation {\em pp} is in the group {\em gp} or not.
wordInStrongGenerators : (PERM S, %)-> L NNI
++ wordInStrongGenerators(p,gp) returns the word for the
++ permutation p in the strong generators of the group {\em gp},
++ represented by the indices of the list, given by {\em strongGenerators}.
wordInGenerators : (PERM S, %)-> L NNI
++ wordInGenerators(p,gp) returns the word for the permutation p
++ in the original generators of the group {\em gp},
++ represented by the indices of the list, given by {\em generators}.
movedPoints : % -> FSET S
++ movedPoints(gp) returns the points moved by the group {\em gp}.
"<" : (%,%) -> B
++ gp1 < gp2 returns true if and only if {\em gp1}
++ is a proper subgroup of {\em gp2}.
"<=" : (%,%) -> B
++ gp1 <= gp2 returns true if and only if {\em gp1}
++ is a subgroup of {\em gp2}.
++ Note: because of a bug in the parser you have to call this
++ function explicitly by {\em gp1 <=$(PERMGRP S) gp2}.
-- (GILT DAS NOCH?)
initializeGroupForWordProblem : % -> Void
++ initializeGroupForWordProblem(gp) initializes the group {\em gp}
++ for the word problem.
++ Notes: it calls the other function of this name with parameters
++ 0 and 1: {\em initializeGroupForWordProblem(gp,0,1)}.
++ Notes: (1) be careful: invoking this routine will destroy the
++ possibly information about your group (but will recompute it again)
++ (2) users need not call this function normally for the soultion of
++ the word problem.
initializeGroupForWordProblem :(%,I,I) -> Void
++ initializeGroupForWordProblem(gp,m,n) initializes the group
++ {\em gp} for the word problem.
++ Notes: (1) with a small integer you get shorter words, but the
++ routine takes longer than the standard routine for longer words.
++ (2) be careful: invoking this routine will destroy the possibly stored
++ information about your group (but will recompute it again).
++ (3) users need not call this function normally for the soultion of
++ the word problem.
private ==> add
-- representation of the object:
Rep := Record ( gens : L PERM S , information : REC2 )
-- import of domains and packages
import Permutation S
import OutputForm
import Symbol
import Void
--first the local variables
sgs : L V NNI := []
baseOfGroup : L NNI := []
sizeOfGroup : NNI := 1
degree : NNI := 0
gporb : L REC := []
out : L L V NNI := []
outword : L L L NNI := []
wordlist : L L NNI := []
basePoint : NNI := 0
newBasePoint : B := true
supp : L S := []
ord : NNI := 1
wordProblem : B := true
--local functions first, signatures:
shortenWord:(L NNI, %)->L NNI
times:(V NNI, V NNI)->V NNI
strip:(V NNI,REC,L V NNI,L L NNI)->REC3
orbitInternal:(%,L S )->L L S
inv: V NNI->V NNI
ranelt:(L V NNI,L L NNI, I)->REC3
testIdentity:V NNI->B
pointList: %->L S
orbitWithSvc:(L V NNI ,NNI )->REC
cosetRep:(NNI ,REC ,L V NNI )->REC3
bsgs1:(L V NNI,NNI,L L NNI,I,%,I)->NNI
computeOrbits: I->L NNI
reduceGenerators: I->Void
bsgs:(%, I, I)->NNI
initialize: %->FSET PERM S
knownGroup?: %->Void
subgroup:(%, %)->B
memberInternal:(PERM S, %, B)->REC4
--local functions first, implementations:
shortenWord ( lw : L NNI , gp : % ) : L NNI ==
-- tries to shorten a word in the generators by removing identities
gpgens : L PERM S := coerce gp
orderList : L NNI := [ order gen for gen in gpgens ]
newlw : L NNI := copy lw
for i in 1.. maxIndex orderList repeat
if orderList.i = 1 then
while member?(i,newlw) repeat
-- removing the trivial element
pos := position(i,newlw)
newlw := delete(newlw,pos)
flag : B := true
while flag repeat
actualLength : NNI := (maxIndex newlw) pretend NNI
pointer := actualLength
test := newlw.pointer
anzahl : NNI := 1
flag := false
while pointer > 1 repeat
pointer := ( pointer - 1 )::NNI
if newlw.pointer ~= test then
-- don't get a trivial element, try next
test := newlw.pointer
anzahl := 1
else
anzahl := anzahl + 1
if anzahl = orderList.test then
-- we have an identity, so remove it
for i in (pointer+anzahl)..actualLength repeat
newlw.(i-anzahl) := newlw.i
newlw := first(newlw, (actualLength - anzahl) :: NNI)
flag := true
pointer := 1
newlw
times ( p : V NNI , q : V NNI ) : V NNI ==
-- internal multiplication of permutations
[ qelt(p,qelt(q,i)) for i in 1..degree ]
strip(element:V NNI,orbit:REC,group:L V NNI,words:L L NNI) : REC3 ==
-- strip an element into the stabilizer
actelt := element
schreierVector := orbit.svc
point := orbit.orb.1
outlist := nil()$(L NNI)
entryLessZero : B := false
while ^entryLessZero repeat
entry := schreierVector.(actelt.point)
entryLessZero := (entry < 0)
if ^entryLessZero then
actelt := times(group.entry, actelt)
if wordProblem then outlist := append ( words.(entry::NNI) , outlist )
[ actelt , reverse outlist ]
orbitInternal ( gp : % , startList : L S ) : L L S ==
orbitList : L L S := [ startList ]
pos : I := 1
while not zero? pos repeat
gpset : L PERM S := gp.gens
for gen in gpset repeat
newList := nil()$(L S)
workList := orbitList.pos
for j in #workList..1 by -1 repeat
newList := cons ( eval ( gen , workList.j ) , newList )
if ^member?( newList , orbitList ) then
orbitList := cons ( newList , orbitList )
pos := pos + 1
pos := pos - 1
reverse orbitList
inv ( p : V NNI ) : V NNI ==
-- internal inverse of a permutation
q : V NNI := new(degree,0)$(V NNI)
for i in 1..degree repeat q.(qelt(p,i)) := i
q
ranelt ( group : L V NNI , word : L L NNI , maxLoops : I ) : REC3 ==
-- generate a "random" element
numberOfGenerators := # group
randomInteger : I := 1 + (random()$Integer rem numberOfGenerators)
randomElement : V NNI := group.randomInteger
words := nil()$(L NNI)
if wordProblem then words := word.(randomInteger::NNI)
if maxLoops > 0 then
numberOfLoops : I := 1 + (random()$Integer rem maxLoops)
else
numberOfLoops : I := maxLoops
while numberOfLoops > 0 repeat
randomInteger : I := 1 + (random()$Integer rem numberOfGenerators)
randomElement := times ( group.randomInteger , randomElement )
if wordProblem then words := append ( word.(randomInteger::NNI) , words)
numberOfLoops := numberOfLoops - 1
[ randomElement , words ]
testIdentity ( p : V NNI ) : B ==
-- internal test for identity
for i in 1..degree repeat qelt(p,i) ~= i => return false
true
pointList(group : %) : L S ==
support : FSET S := brace() -- empty set !!
for perm in group.gens repeat
support := union(support, movedPoints perm)
parts support
orbitWithSvc ( group : L V NNI , point : NNI ) : REC ==
-- compute orbit with Schreier vector, "-2" means not in the orbit,
-- "-1" means starting point, the PI correspond to generators
newGroup := nil()$(L V NNI)
for el in group repeat
newGroup := cons ( inv el , newGroup )
newGroup := reverse newGroup
orbit : L NNI := [ point ]
schreierVector : V I := new ( degree , -2 )
schreierVector.point := -1
position : I := 1
while not zero? position repeat
for i in 1..#newGroup repeat
newPoint := orbit.position
newPoint := newGroup.i.newPoint
if ^ member? ( newPoint , orbit ) then
orbit := cons ( newPoint , orbit )
position := position + 1
schreierVector.newPoint := i
position := position - 1
[ reverse orbit , schreierVector ]
cosetRep ( point : NNI , o : REC , group : L V NNI ) : REC3 ==
ppt := point
xelt : V NNI := [ n for n in 1..degree ]
word := nil()$(L NNI)
oorb := o.orb
osvc := o.svc
while degree > 0 repeat
p := osvc.ppt
p < 0 => return [ xelt , word ]
x := group.p
xelt := times ( x , xelt )
if wordProblem then word := append ( wordlist.p , word )
ppt := x.ppt
bsgs1 (group:L V NNI,number1:NNI,words:L L NNI,maxLoops:I,gp:%,diff:I)_
: NNI ==
-- try to get a good approximation for the strong generators and base
ort: REC
k1: NNI
for i in number1..degree repeat
ort := orbitWithSvc ( group , i )
k := ort.orb
k1 := # k
if k1 ~= 1 then leave
gpsgs := nil()$(L V NNI)
words2 := nil()$(L L NNI)
gplength : NNI := #group
for jj in 1..gplength repeat if (group.jj).i ~= i then leave
for k in 1..gplength repeat
el2 := group.k
if el2.i ~= i then
gpsgs := cons ( el2 , gpsgs )
if wordProblem then words2 := cons ( words.k , words2 )
else
gpsgs := cons ( times ( group.jj , el2 ) , gpsgs )
if wordProblem _
then words2 := cons ( append ( words.jj , words.k ) , words2 )
group2 := nil()$(L V NNI)
words3 := nil()$(L L NNI)
j : I := 15
while j > 0 repeat
-- find generators for the stabilizer
ran := ranelt ( group , words , maxLoops )
str := strip ( ran.elt , ort , group , words )
el2 := str.elt
if ^ testIdentity el2 then
if ^ member?(el2,group2) then
group2 := cons ( el2 , group2 )
if wordProblem then
help : L NNI := append ( reverse str.lst , ran.lst )
help := shortenWord ( help , gp )
words3 := cons ( help , words3 )
j := j - 2
j := j - 1
-- this is for word length control
if wordProblem then maxLoops := maxLoops - diff
if ( null group2 ) or ( maxLoops < 0 ) then
sizeOfGroup := k1
baseOfGroup := [ i ]
out := [ gpsgs ]
outword := [ words2 ]
return sizeOfGroup
k2 := bsgs1 ( group2 , i + 1 , words3 , maxLoops , gp , diff )
sizeOfGroup := k1 * k2
out := append ( out , [ gpsgs ] )
outword := append ( outword , [ words2 ] )
baseOfGroup := cons ( i , baseOfGroup )
sizeOfGroup
computeOrbits ( kkk : I ) : L NNI ==
-- compute the orbits for the stabilizers
sgs := nil()
orbitLength := nil()$(L NNI)
gporb := nil()
for i in 1..#baseOfGroup repeat
sgs := append ( sgs , out.i )
pt := #baseOfGroup - i + 1
obs := orbitWithSvc ( sgs , baseOfGroup.pt )
orbitLength := cons ( #obs.orb , orbitLength )
gporb := cons ( obs , gporb )
gporb := reverse gporb
reverse orbitLength
reduceGenerators ( kkk : I ) : Void ==
-- try to reduce number of strong generators
orbitLength := computeOrbits ( kkk )
sgs := nil()
wordlist := nil()
for i in 1..(kkk-1) repeat
sgs := append ( sgs , out.i )
if wordProblem then wordlist := append ( wordlist , outword.i )
removedGenerator := false
baseLength : NNI := #baseOfGroup
for nnn in kkk..(baseLength-1) repeat
sgs := append ( sgs , out.nnn )
if wordProblem then wordlist := append ( wordlist , outword.nnn )
pt := baseLength - nnn + 1
obs := orbitWithSvc ( sgs , baseOfGroup.pt )
i := 1
while not ( i > # out.nnn ) repeat
pos := position ( out.nnn.i , sgs )
sgs2 := delete(sgs, pos)
obs2 := orbitWithSvc ( sgs2 , baseOfGroup.pt )
if # obs2.orb = orbitLength.nnn then
test := true
for j in (nnn+1)..(baseLength-1) repeat
pt2 := baseLength - j + 1
sgs2 := append ( sgs2 , out.j )
obs2 := orbitWithSvc ( sgs2 , baseOfGroup.pt2 )
if # obs2.orb ~= orbitLength.j then
test := false
leave
if test then
removedGenerator := true
sgs := delete (sgs, pos)
if wordProblem then wordlist := delete(wordlist, pos)
out.nnn := delete (out.nnn, i)
if wordProblem then _
outword.nnn := delete(outword.nnn, i )
else
i := i + 1
else
i := i + 1
if removedGenerator then orbitLength := computeOrbits ( kkk )
void()
bsgs ( group : % ,maxLoops : I , diff : I ) : NNI ==
-- the MOST IMPORTANT part of the package
supp := pointList group
degree := # supp
if degree = 0 then
sizeOfGroup := 1
sgs := [ [ 0 ] ]
baseOfGroup := nil()
gporb := nil()
return sizeOfGroup
newGroup := nil()$(L V NNI)
gp : L PERM S := group.gens
words := nil()$(L L NNI)
for ggg in 1..#gp repeat
q := new(degree,0)$(V NNI)
for i in 1..degree repeat
newEl := eval ( gp.ggg , supp.i )
pos2 := position ( newEl , supp )
q.i := pos2 pretend NNI
newGroup := cons ( q , newGroup )
if wordProblem then words := cons(list ggg, words)
if maxLoops < 1 then
-- try to get the (approximate) base length
if zero? (# ((group.information).gpbase)) then
wordProblem := false
k := bsgs1 ( newGroup , 1 , words , 20 , group , 0 )
wordProblem := true
maxLoops := (# baseOfGroup) - 1
else
maxLoops := (# ((group.information).gpbase)) - 1
k := bsgs1 ( newGroup , 1 , words , maxLoops , group , diff )
kkk : I := 1
newGroup := reverse newGroup
noAnswer : B := true
z: V NNI
while noAnswer repeat
reduceGenerators kkk
-- *** Here is former "bsgs2" *** --
-- test whether we have a base and a strong generating set
sgs := nil()
wordlist := nil()
for i in 1..(kkk-1) repeat
sgs := append ( sgs , out.i )
if wordProblem then wordlist := append ( wordlist , outword.i )
noresult : B := true
word3: L NNI
word: L NNI
for i in kkk..#baseOfGroup while noresult repeat
sgs := append ( sgs , out.i )
if wordProblem then wordlist := append ( wordlist , outword.i )
gporbi := gporb.i
for pt in gporbi.orb while noresult repeat
ppp := cosetRep ( pt , gporbi , sgs )
y1 := inv ppp.elt
word3 := ppp.lst
for jjj in 1..#sgs while noresult repeat
word := nil()$(L NNI)
z := times ( sgs.jjj , y1 )
if wordProblem then word := append ( wordlist.jjj , word )
ppp := cosetRep ( (sgs.jjj).pt , gporbi , sgs )
z := times ( ppp.elt , z )
if wordProblem then word := append ( ppp.lst , word )
newBasePoint := false
for j in (i-1)..1 by -1 while noresult repeat
s := gporb.j.svc
p := gporb.j.orb.1
while ( degree > 0 ) and noresult repeat
entry := s.(z.p)
if entry < 0 then
if entry = -1 then leave
basePoint := j::NNI
noresult := false
else
ee := sgs.entry
z := times ( ee , z )
if wordProblem then word := append ( wordlist.entry , word )
if noresult then
basePoint := 1
newBasePoint := true
noresult := testIdentity z
noAnswer := not (testIdentity z)
if noAnswer then
-- we have missed something
word2 := nil()$(L NNI)
if wordProblem then
for wd in word3 repeat
ttt := newGroup.wd
while not (testIdentity ttt) repeat
word2 := cons ( wd , word2 )
ttt := times ( ttt , newGroup.wd )
word := append ( word , word2 )
word := shortenWord ( word , group )
if newBasePoint then
for i in 1..degree repeat
if z.i ~= i then
baseOfGroup := append ( baseOfGroup , [ i ] )
leave
out := cons (list z, out )
if wordProblem then outword := cons (list word , outword )
else
out.basePoint := cons ( z , out.basePoint )
if wordProblem then outword.basePoint := cons(word ,outword.basePoint )
kkk := basePoint
sizeOfGroup := 1
for j in 1..#baseOfGroup repeat
sizeOfGroup := sizeOfGroup * # gporb.j.orb
sizeOfGroup
initialize ( group : % ) : FSET PERM S ==
group2 := brace()$(FSET PERM S)
gp : L PERM S := group.gens
for gen in gp repeat
if degree gen > 0 then insert_!(gen, group2)
group2
knownGroup? (gp : %) : Void ==
-- do we know the group already?
result := gp.information
if result.order = 0 then
wordProblem := false
ord := bsgs ( gp , 20 , 0 )
result := [ ord , sgs , baseOfGroup , gporb , supp , [] ]
gp.information := result
else
ord := result.order
sgs := result.sgset
baseOfGroup := result.gpbase
gporb := result.orbs
supp := result.mp
wordlist := result.wd
void
subgroup ( gp1 : % , gp2 : % ) : B ==
gpset1 := initialize gp1
gpset2 := initialize gp2
empty? difference (gpset1, gpset2) => true
for el in parts gpset1 repeat
not member? (el, gp2) => return false
true
memberInternal ( p : PERM S , gp : % , flag : B ) : REC4 ==
-- internal membership testing
supp := pointList gp
outlist := nil()$(L NNI)
mP : L S := parts movedPoints p
for x in mP repeat
not member? (x, supp) => return [ false , nil()$(L NNI) ]
if flag then
member? ( p , gp.gens ) => return [ true , nil()$(L NNI) ]
knownGroup? gp
else
result := gp.information
if #(result.wd) = 0 then
initializeGroupForWordProblem gp
else
ord := result.order
sgs := result.sgset
baseOfGroup := result.gpbase
gporb := result.orbs
supp := result.mp
wordlist := result.wd
degree := # supp
pp := new(degree,0)$(V NNI)
for i in 1..degree repeat
el := eval ( p , supp.i )
pos := position ( el , supp )
pp.i := pos::NNI
words := nil()$(L L NNI)
if wordProblem then
for i in 1..#sgs repeat
lw : L NNI := [ (#sgs - i + 1)::NNI ]
words := cons ( lw , words )
for i in #baseOfGroup..1 by -1 repeat
str := strip ( pp , gporb.i , sgs , words )
pp := str.elt
if wordProblem then outlist := append ( outlist , str.lst )
[ testIdentity pp , reverse outlist ]
--now the exported functions
coerce ( gp : % ) : L PERM S == gp.gens
generators ( gp : % ) : L PERM S == gp.gens
strongGenerators ( group ) ==
knownGroup? group
degree := # supp
strongGens := nil()$(L PERM S)
for i in sgs repeat
pairs := nil()$(L L S)
for j in 1..degree repeat
pairs := cons ( [ supp.j , supp.(i.j) ] , pairs )
strongGens := cons ( coerceListOfPairs pairs , strongGens )
reverse strongGens
elt ( gp , i ) == (gp.gens).i
movedPoints ( gp ) == brace pointList gp
random ( group , maximalNumberOfFactors ) ==
maximalNumberOfFactors < 1 => 1$(PERM S)
gp : L PERM S := group.gens
numberOfGenerators := # gp
randomInteger : I := 1 + (random()$Integer rem numberOfGenerators)
randomElement := gp.randomInteger
numberOfLoops : I := 1 + (random()$Integer rem maximalNumberOfFactors)
while numberOfLoops > 0 repeat
randomInteger : I := 1 + (random()$Integer rem numberOfGenerators)
randomElement := gp.randomInteger * randomElement
numberOfLoops := numberOfLoops - 1
randomElement
random ( group ) == random ( group , 20 )
order ( group ) ==
knownGroup? group
ord
degree ( group ) == # pointList group
base ( group ) ==
knownGroup? group
groupBase := nil()$(L S)
for i in baseOfGroup repeat
groupBase := cons ( supp.i , groupBase )
reverse groupBase
wordsForStrongGenerators ( group ) ==
knownGroup? group
wordlist
coerce ( gp : L PERM S ) : % ==
result : REC2 := [ 0 , [] , [] , [] , [] , [] ]
group := [ gp , result ]
permutationGroup ( gp : L PERM S ) : % ==
result : REC2 := [ 0 , [] , [] , [] , [] , [] ]
group := [ gp , result ]
coerce(group: %) : OUT ==
outList := nil()$(L OUT)
gp : L PERM S := group.gens
for i in (maxIndex gp)..1 by -1 repeat
outList := cons(coerce gp.i, outList)
postfix(outputForm(">":SYM),postfix(commaSeparate outList,outputForm("<":SYM)))
orbit ( gp : % , el : S ) : FSET S ==
elList : L S := [ el ]
outList := orbitInternal ( gp , elList )
outSet := brace()$(FSET S)
for i in 1..#outList repeat
insert_! ( outList.i.1 , outSet )
outSet
orbits ( gp ) ==
spp := movedPoints gp
orbits := nil()$(L FSET S)
while cardinality spp > 0 repeat
el := extract_! spp
orbitSet := orbit ( gp , el )
orbits := cons ( orbitSet , orbits )
spp := difference ( spp , orbitSet )
brace orbits
member? (p, gp) ==
wordProblem := false
mi := memberInternal ( p , gp , true )
mi.bool
wordInStrongGenerators (p, gp ) ==
mi := memberInternal ( inv p , gp , false )
not mi.bool => error "p is not an element of gp"
mi.lst
wordInGenerators (p, gp) ==
lll : L NNI := wordInStrongGenerators (p, gp)
outlist := nil()$(L NNI)
for wd in lll repeat
outlist := append ( outlist , wordlist.wd )
shortenWord ( outlist , gp )
gp1 < gp2 ==
not empty? difference ( movedPoints gp1 , movedPoints gp2 ) => false
not subgroup ( gp1 , gp2 ) => false
order gp1 = order gp2 => false
true
gp1 <= gp2 ==
not empty? difference ( movedPoints gp1 , movedPoints gp2 ) => false
subgroup ( gp1 , gp2 )
gp1 = gp2 ==
movedPoints gp1 ~= movedPoints gp2 => false
if #(gp1.gens) <= #(gp2.gens) then
not subgroup ( gp1 , gp2 ) => return false
else
not subgroup ( gp2 , gp1 ) => return false
order gp1 = order gp2 => true
false
orbit ( gp : % , startSet : FSET S ) : FSET FSET S ==
startList : L S := parts startSet
outList := orbitInternal ( gp , startList )
outSet := brace()$(FSET FSET S)
for i in 1..#outList repeat
newSet : FSET S := brace outList.i
insert_! ( newSet , outSet )
outSet
orbit ( gp : % , startList : L S ) : FSET L S ==
brace orbitInternal(gp, startList)
initializeGroupForWordProblem ( gp , maxLoops , diff ) ==
wordProblem := true
ord := bsgs ( gp , maxLoops , diff )
gp.information := [ ord , sgs , baseOfGroup , gporb , supp , wordlist ]
void
initializeGroupForWordProblem ( gp ) == initializeGroupForWordProblem ( gp , 0 , 1 )
@
\section{package PGE PermutationGroupExamples}
<<package PGE PermutationGroupExamples>>=
)abbrev package PGE PermutationGroupExamples
++ Authors: M. Weller, G. Schneider, J. Grabmeier
++ Date Created: 20 February 1990
++ Date Last Updated: 09 June 1990
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson:
++ Atlas of Finite Groups, Oxford, Clarendon Press, 1987
++ Description:
++ PermutationGroupExamples provides permutation groups for
++ some classes of groups: symmetric, alternating, dihedral, cyclic,
++ direct products of cyclic, which are in fact the finite abelian groups
++ of symmetric groups called Young subgroups.
++ Furthermore, Rubik's group as permutation group of 48 integers and a list
++ of sporadic simple groups derived from the atlas of finite groups.
PermutationGroupExamples():public == private where
L ==> List
I ==> Integer
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
PERM ==> Permutation
PERMGRP ==> PermutationGroup
public ==> with
symmetricGroup: PI -> PERMGRP I
++ symmetricGroup(n) constructs the symmetric group {\em Sn}
++ acting on the integers 1,...,n, generators are the
++ {\em n}-cycle {\em (1,...,n)} and the 2-cycle {\em (1,2)}.
symmetricGroup: L I -> PERMGRP I
++ symmetricGroup(li) constructs the symmetric group acting on
++ the integers in the list {\em li}, generators are the
++ cycle given by {\em li} and the 2-cycle {\em (li.1,li.2)}.
++ Note: duplicates in the list will be removed.
alternatingGroup: PI -> PERMGRP I
++ alternatingGroup(n) constructs the alternating group {\em An}
++ acting on the integers 1,...,n, generators are in general the
++ {\em n-2}-cycle {\em (3,...,n)} and the 3-cycle {\em (1,2,3)}
++ if n is odd and the product of the 2-cycle {\em (1,2)} with
++ {\em n-2}-cycle {\em (3,...,n)} and the 3-cycle {\em (1,2,3)}
++ if n is even.
alternatingGroup: L I -> PERMGRP I
++ alternatingGroup(li) constructs the alternating group acting
++ on the integers in the list {\em li}, generators are in general the
++ {\em n-2}-cycle {\em (li.3,...,li.n)} and the 3-cycle
++ {\em (li.1,li.2,li.3)}, if n is odd and
++ product of the 2-cycle {\em (li.1,li.2)} with
++ {\em n-2}-cycle {\em (li.3,...,li.n)} and the 3-cycle
++ {\em (li.1,li.2,li.3)}, if n is even.
++ Note: duplicates in the list will be removed.
abelianGroup: L PI -> PERMGRP I
++ abelianGroup([n1,...,nk]) constructs the abelian group that
++ is the direct product of cyclic groups with order {\em ni}.
cyclicGroup: PI -> PERMGRP I
++ cyclicGroup(n) constructs the cyclic group of order n acting
++ on the integers 1,...,n.
cyclicGroup: L I -> PERMGRP I
++ cyclicGroup([i1,...,ik]) constructs the cyclic group of
++ order k acting on the integers {\em i1},...,{\em ik}.
++ Note: duplicates in the list will be removed.
dihedralGroup: PI -> PERMGRP I
++ dihedralGroup(n) constructs the dihedral group of order 2n
++ acting on integers 1,...,N.
dihedralGroup: L I -> PERMGRP I
++ dihedralGroup([i1,...,ik]) constructs the dihedral group of
++ order 2k acting on the integers out of {\em i1},...,{\em ik}.
++ Note: duplicates in the list will be removed.
mathieu11: L I -> PERMGRP I
++ mathieu11(li) constructs the mathieu group acting on the 11
++ integers given in the list {\em li}.
++ Note: duplicates in the list will be removed.
++ error, if {\em li} has less or more than 11 different entries.
mathieu11: () -> PERMGRP I
++ mathieu11 constructs the mathieu group acting on the
++ integers 1,...,11.
mathieu12: L I -> PERMGRP I
++ mathieu12(li) constructs the mathieu group acting on the 12
++ integers given in the list {\em li}.
++ Note: duplicates in the list will be removed
++ Error: if {\em li} has less or more than 12 different entries.
mathieu12: () -> PERMGRP I
++ mathieu12 constructs the mathieu group acting on the
++ integers 1,...,12.
mathieu22: L I -> PERMGRP I
++ mathieu22(li) constructs the mathieu group acting on the 22
++ integers given in the list {\em li}.
++ Note: duplicates in the list will be removed.
++ Error: if {\em li} has less or more than 22 different entries.
mathieu22: () -> PERMGRP I
++ mathieu22 constructs the mathieu group acting on the
++ integers 1,...,22.
mathieu23: L I -> PERMGRP I
++ mathieu23(li) constructs the mathieu group acting on the 23
++ integers given in the list {\em li}.
++ Note: duplicates in the list will be removed.
++ Error: if {\em li} has less or more than 23 different entries.
mathieu23: () -> PERMGRP I
++ mathieu23 constructs the mathieu group acting on the
++ integers 1,...,23.
mathieu24: L I -> PERMGRP I
++ mathieu24(li) constructs the mathieu group acting on the 24
++ integers given in the list {\em li}.
++ Note: duplicates in the list will be removed.
++ Error: if {\em li} has less or more than 24 different entries.
mathieu24: () -> PERMGRP I
++ mathieu24 constructs the mathieu group acting on the
++ integers 1,...,24.
janko2: L I -> PERMGRP I
++ janko2(li) constructs the janko group acting on the 100
++ integers given in the list {\em li}.
++ Note: duplicates in the list will be removed.
++ Error: if {\em li} has less or more than 100 different entries
janko2: () -> PERMGRP I
++ janko2 constructs the janko group acting on the
++ integers 1,...,100.
rubiksGroup: () -> PERMGRP I
++ rubiksGroup constructs the permutation group representing
++ Rubic's Cube acting on integers {\em 10*i+j} for
++ {\em 1 <= i <= 6}, {\em 1 <= j <= 8}.
++ The faces of Rubik's Cube are labelled in the obvious way
++ Front, Right, Up, Down, Left, Back and numbered from 1 to 6
++ in this given ordering, the pieces on each face
++ (except the unmoveable center piece) are clockwise numbered
++ from 1 to 8 starting with the piece in the upper left
++ corner. The moves of the cube are represented as permutations
++ on these pieces, represented as a two digit
++ integer {\em ij} where i is the numer of theface (1 to 6)
++ and j is the number of the piece on this face.
++ The remaining ambiguities are resolved by looking
++ at the 6 generators, which represent a 90 degree turns of the
++ faces, or from the following pictorial description.
++ Permutation group representing Rubic's Cube acting on integers
++ 10*i+j for 1 <= i <= 6, 1 <= j <=8.
++
++ \begin{verbatim}
++ Rubik's Cube: +-----+ +-- B where: marks Side # :
++ / U /|/
++ / / | F(ront) <-> 1
++ L --> +-----+ R| R(ight) <-> 2
++ | | + U(p) <-> 3
++ | F | / D(own) <-> 4
++ | |/ L(eft) <-> 5
++ +-----+ B(ack) <-> 6
++ ^
++ |
++ D
++
++ The Cube's surface:
++ The pieces on each side
++ +---+ (except the unmoveable center
++ |567| piece) are clockwise numbered
++ |4U8| from 1 to 8 starting with the
++ |321| piece in the upper left
++ +---+---+---+ corner (see figure on the
++ |781|123|345| left). The moves of the cube
++ |6L2|8F4|2R6| are represented as
++ |543|765|187| permutations on these pieces.
++ +---+---+---+ Each of the pieces is
++ |123| represented as a two digit
++ |8D4| integer ij where i is the
++ |765| # of the side ( 1 to 6 for
++ +---+ F to B (see table above ))
++ |567| and j is the # of the piece.
++ |4B8|
++ |321|
++ +---+
++ \end{verbatim}
youngGroup: L I -> PERMGRP I
++ youngGroup([n1,...,nk]) constructs the direct product of the
++ symmetric groups {\em Sn1},...,{\em Snk}.
youngGroup: Partition -> PERMGRP I
++ youngGroup(lambda) constructs the direct product of the symmetric
++ groups given by the parts of the partition {\em lambda}.
private ==> add
-- import the permutation and permutation group domains:
import PERM I
import PERMGRP I
-- import the needed map function:
import ListFunctions2(L L I,PERM I)
-- the internal functions:
llli2gp(l:L L L I):PERMGRP I ==
--++ Converts an list of permutations each represented by a list
--++ of cycles ( each of them represented as a list of Integers )
--++ to the permutation group generated by these permutations.
(map(cycles,l))::PERMGRP I
li1n(n:I):L I ==
--++ constructs the list of integers from 1 to n
[i for i in 1..n]
-- definition of the exported functions:
youngGroup(l:L I):PERMGRP I ==
gens:= nil()$(L L L I)
element:I:= 1
for n in l | n > 1 repeat
gens:=cons(list [i for i in element..(element+n-1)], gens)
if n >= 3 then gens := cons([[element,element+1]],gens)
element:=element+n
llli2gp
#gens = 0 => [[[1]]]
gens
youngGroup(lambda : Partition):PERMGRP I ==
youngGroup(convert(lambda)$Partition)
rubiksGroup():PERMGRP I ==
-- each generator represents a 90 degree turn of the appropriate
-- side.
f:L L I:=
[[11,13,15,17],[12,14,16,18],[51,31,21,41],[53,33,23,43],[52,32,22,42]]
r:L L I:=
[[21,23,25,27],[22,24,26,28],[13,37,67,43],[15,31,61,45],[14,38,68,44]]
u:L L I:=
[[31,33,35,37],[32,34,36,38],[13,51,63,25],[11,57,61,23],[12,58,62,24]]
d:L L I:=
[[41,43,45,47],[42,44,46,48],[17,21,67,55],[15,27,65,53],[16,28,66,54]]
l:L L I:=
[[51,53,55,57],[52,54,56,58],[11,41,65,35],[17,47,63,33],[18,48,64,34]]
b:L L I:=
[[61,63,65,67],[62,64,66,68],[45,25,35,55],[47,27,37,57],[46,26,36,56]]
llli2gp [f,r,u,d,l,b]
mathieu11(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ~= 11 => error "Exactly 11 integers for mathieu11 needed !"
a:L L I:=[[l.1,l.10],[l.2,l.8],[l.3,l.11],[l.5,l.7]]
llli2gp [a,[[l.1,l.4,l.7,l.6],[l.2,l.11,l.10,l.9]]]
mathieu11():PERMGRP I == mathieu11 li1n 11
mathieu12(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ~= 12 => error "Exactly 12 integers for mathieu12 needed !"
a:L L I:=
[[l.1,l.2,l.3,l.4,l.5,l.6,l.7,l.8,l.9,l.10,l.11]]
llli2gp [a,[[l.1,l.6,l.5,l.8,l.3,l.7,l.4,l.2,l.9,l.10],[l.11,l.12]]]
mathieu12():PERMGRP I == mathieu12 li1n 12
mathieu22(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ~= 22 => error "Exactly 22 integers for mathieu22 needed !"
a:L L I:=[[l.1,l.2,l.4,l.8,l.16,l.9,l.18,l.13,l.3,l.6,l.12], _
[l.5,l.10,l.20,l.17,l.11,l.22,l.21,l.19,l.15,l.7,l.14]]
b:L L I:= [[l.1,l.2,l.6,l.18],[l.3,l.15],[l.5,l.8,l.21,l.13], _
[l.7,l.9,l.20,l.12],[l.10,l.16],[l.11,l.19,l.14,l.22]]
llli2gp [a,b]
mathieu22():PERMGRP I == mathieu22 li1n 22
mathieu23(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ~= 23 => error "Exactly 23 integers for mathieu23 needed !"
a:L L I:= [[l.1,l.2,l.3,l.4,l.5,l.6,l.7,l.8,l.9,l.10,l.11,l.12,l.13,l.14,_
l.15,l.16,l.17,l.18,l.19,l.20,l.21,l.22,l.23]]
b:L L I:= [[l.2,l.16,l.9,l.6,l.8],[l.3,l.12,l.13,l.18,l.4], _
[l.7,l.17,l.10,l.11,l.22],[l.14,l.19,l.21,l.20,l.15]]
llli2gp [a,b]
mathieu23():PERMGRP I == mathieu23 li1n 23
mathieu24(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ~= 24 => error "Exactly 24 integers for mathieu24 needed !"
a:L L I:= [[l.1,l.16,l.10,l.22,l.24],[l.2,l.12,l.18,l.21,l.7], _
[l.4,l.5,l.8,l.6,l.17],[l.9,l.11,l.13,l.19,l.15]]
b:L L I:= [[l.1,l.22,l.13,l.14,l.6,l.20,l.3,l.21,l.8,l.11],[l.2,l.10], _
[l.4,l.15,l.18,l.17,l.16,l.5,l.9,l.19,l.12,l.7],[l.23,l.24]]
llli2gp [a,b]
mathieu24():PERMGRP I == mathieu24 li1n 24
janko2(l:L I):PERMGRP I ==
-- permutations derived from the ATLAS
l:=removeDuplicates l
#l ~= 100 => error "Exactly 100 integers for janko2 needed !"
a:L L I:=[ _
[l.2,l.3,l.4,l.5,l.6,l.7,l.8], _
[l.9,l.10,l.11,l.12,l.13,l.14,l.15], _
[l.16,l.17,l.18,l.19,l.20,l.21,l.22], _
[l.23,l.24,l.25,l.26,l.27,l.28,l.29], _
[l.30,l.31,l.32,l.33,l.34,l.35,l.36], _
[l.37,l.38,l.39,l.40,l.41,l.42,l.43], _
[l.44,l.45,l.46,l.47,l.48,l.49,l.50], _
[l.51,l.52,l.53,l.54,l.55,l.56,l.57], _
[l.58,l.59,l.60,l.61,l.62,l.63,l.64], _
[l.65,l.66,l.67,l.68,l.69,l.70,l.71], _
[l.72,l.73,l.74,l.75,l.76,l.77,l.78], _
[l.79,l.80,l.81,l.82,l.83,l.84,l.85], _
[l.86,l.87,l.88,l.89,l.90,l.91,l.92], _
[l.93,l.94,l.95,l.96,l.97,l.98,l.99] ]
b:L L I:=[
[l.1,l.74,l.83,l.21,l.36,l.77,l.44,l.80,l.64,l.2,l.34,l.75,l.48,l.17,l.100],_
[l.3,l.15,l.31,l.52,l.19,l.11,l.73,l.79,l.26,l.56,l.41,l.99,l.39,l.84,l.90],_
[l.4,l.57,l.86,l.63,l.85,l.95,l.82,l.97,l.98,l.81,l.8,l.69,l.38,l.43,l.58],_
[l.5,l.66,l.49,l.59,l.61],_
[l.6,l.68,l.89,l.94,l.92,l.20,l.13,l.54,l.24,l.51,l.87,l.27,l.76,l.23,l.67],_
[l.7,l.72,l.22,l.35,l.30,l.70,l.47,l.62,l.45,l.46,l.40,l.28,l.65,l.93,l.42],_
[l.9,l.71,l.37,l.91,l.18,l.55,l.96,l.60,l.16,l.53,l.50,l.25,l.32,l.14,l.33],_
[l.10,l.78,l.88,l.29,l.12] ]
llli2gp [a,b]
janko2():PERMGRP I == janko2 li1n 100
abelianGroup(l:L PI):PERMGRP I ==
gens:= nil()$(L L L I)
element:I:= 1
for n in l | n > 1 repeat
gens:=cons( list [i for i in element..(element+n-1) ], gens )
element:=element+n
llli2gp
#gens = 0 => [[[1]]]
gens
alternatingGroup(l:L I):PERMGRP I ==
l:=removeDuplicates l
#l = 0 =>
error "Cannot construct alternating group on empty set"
#l < 3 => llli2gp [[[l.1]]]
#l = 3 => llli2gp [[[l.1,l.2,l.3]]]
tmp:= [l.i for i in 3..(#l)]
gens:L L L I:=[[tmp],[[l.1,l.2,l.3]]]
odd?(#l) => llli2gp gens
gens.1 := cons([l.1,l.2],gens.1)
llli2gp gens
alternatingGroup(n:PI):PERMGRP I == alternatingGroup li1n n
symmetricGroup(l:L I):PERMGRP I ==
l:=removeDuplicates l
#l = 0 => error "Cannot construct symmetric group on empty set !"
#l < 3 => llli2gp [[l]]
llli2gp [[l],[[l.1,l.2]]]
symmetricGroup(n:PI):PERMGRP I == symmetricGroup li1n n
cyclicGroup(l:L I):PERMGRP I ==
l:=removeDuplicates l
#l = 0 => error "Cannot construct cyclic group on empty set"
llli2gp [[l]]
cyclicGroup(n:PI):PERMGRP I == cyclicGroup li1n n
dihedralGroup(l:L I):PERMGRP I ==
l:=removeDuplicates l
#l < 3 => error "in dihedralGroup: Minimum of 3 elements needed !"
tmp := [[l.i, l.(#l-i+1) ] for i in 1..(#l quo 2)]
llli2gp [ [ l ], tmp ]
dihedralGroup(n:PI):PERMGRP I ==
n = 1 => symmetricGroup (2::PI)
n = 2 => llli2gp [[[1,2]],[[3,4]]]
dihedralGroup li1n n
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<domain PERMGRP PermutationGroup>>
<<package PGE PermutationGroupExamples>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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