Bruteforce algorithm for creating all elements of a group presented by generators and relations. This current version runs very very slow even for smallest groups.
For example, it takes ~2000ms for generating Symm4 group of permutations.
WordGroup.Generate("a2", "b2", "abab"); // Klein
WordGroup.Generate("a3", "b2", "aba-1b-1"); // C6
WordGroup.Generate("a2", "b2", "ababab"); // S3
WordGroup.Generate("a4", "a2b-2", "b-1aba"); // H8
WordGroup.Generate("a3", "b6", "ab = ba"); // C3 x C6
WordGroup.Generate("a2", "b2", "c2", "bcbcbc", "acacac", "abab"); // S4
will produce
G = { (), a, b, ab }
() => ()
a => a
b => b
ab => ab
Order = 4
Is Group : True
Is Abelian : True
() | a b ab
-------------
a | () ab b
b | ab () a
ab | b a ()
Repr : ()
() => ()
a2 => aa
b2 => bb
a-2 => AA
b-2 => BB
abab => abab
baba => baba
Repr : a
a => a
a-1 => A
bab => bab
Repr : b
b => b
b-1 => B
aba => aba
Repr : ab
ab => ab
ba => ba
Total Words : 15
Total Time : 29 ms; Total Created Words : 1867
and
G = { (), a, b, a-1, ab, ba-1 }
() => ()
a => a
b => b
a-1 => A
ab => ab
ba-1 => bA
Order = 6
Is Group : True
Is Abelian : True
() | a b A ab bA
-------------------
a | A ab () bA b
b | ab () bA a A
A | () bA a b ab
ab | bA a b A ()
bA | b A ab () a
Repr : ()
() => ()
b2 => bb
b-2 => BB
a3 => aaa
a-3 => AAA
aba-1b => abAb
baba-1 => babA
ba-1ba => bAba
a-1bab => Abab
Repr : a
a => a
a-2 => AA
bab => bab
Repr : b
b => b
b-1 => B
aba-1 => abA
a-1ba => Aba
Repr : A
a-1 => A
a2 => aa
ba-1b => bAb
Repr : ab
ab => ab
ba => ba
Repr : bA
ba-1 => bA
a-1b => Ab
Total Words : 23
Total Time : 71 ms; Total Created Words : 4001
and
G = { (), a, b, ab, ba, aba }
() => ()
a => a
b => b
ab => ab
ba => ba
aba => aba
Order = 6
Is Group : True
Is Abelian : False
() | a b ab ba aba
-------------------------
a | () ab b aba ba
b | ba () aba a ab
ab | aba a ba () b
ba | b aba () ab a
aba | ab ba a b ()
Repr : ()
() => ()
a2 => aa
b2 => bb
a-2 => AA
b-2 => BB
ababab => ababab
bababa => bababa
Repr : a
a => a
a-1 => A
babab => babab
Repr : b
b => b
b-1 => B
ababa => ababa
Repr : ab
ab => ab
baba => baba
Repr : ba
ba => ba
abab => abab
Repr : aba
aba => aba
bab => bab
Total Words : 19
Total Time : 66 ms; Total Created Words : 3315
and
G = { (), a, b, a-1, b-1, a2, ab, ab-1 }
() => ()
a => a
b => b
a-1 => A
b-1 => B
a2 => aa
ab => ab
ab-1 => aB
Order = 8
Is Group : True
Is Abelian : False
() | a b A B aa ab aB
-------------------------
a | aa ab () aB A B b
b | aB aa ab () B a A
A | () aB aa ab a b B
B | ab () aB aa b A a
aa | A B a b () aB ab
ab | b A B a aB aa ()
aB | B a b A ab () aa
Repr : ()
() => ()
a4 => aaaa
a-4 => AAAA
a2b2 => aabb
a2b-2 => aaBB
b2a2 => bbaa
b-2a2 => BBaa
ab-2a => aBBa
ba2b => baab
a-1b2a-1 => AbbA
b-1a2b-1 => BaaB
abab-1 => abaB
ab-1a-1b-1 => aBAB
baba-1 => babA
ba-1b-1a-1 => bABA
Repr : a
a => a
a-3 => AAA
a-1b2 => Abb
b2a-1 => bbA
ba-1b-1 => bAB
Repr : b
b => b
b-3 => BBB
b-1a2 => Baa
a2b-1 => aaB
aba => aba
ab-1a-1 => aBA
Repr : A
a-1 => A
a3 => aaa
ab-2 => aBB
b-2a => BBa
bab-1 => baB
Repr : B
b-1 => B
b3 => bbb
ba2 => baa
a2b => aab
aba-1 => abA
a-1b-1a-1 => ABA
Repr : aa
a2 => aa
b2 => bb
a-2 => AA
b-2 => BB
Repr : ab
ab => ab
ba-1 => bA
a-1b-1 => AB
b-1a => Ba
b3a => bbba
b-3a-1 => BBBA
Repr : aB
ab-1 => aB
ba => ba
a-1b => Ab
b-1a-1 => BA
ab3 => abbb
Total Words : 52
Total Time : 573 ms; Total Created Words : 34791
and
G = { (), a, b, a-1, b-1, b2, b-2, b3, ab, ab-1, ba-1, a-1b-1, ab2, ab-2, a-1b2, a-1b-2, ab3, a-1b3 }
() => ()
a => a
b => b
a-1 => A
b-1 => B
b2 => bb
b-2 => BB
b3 => bbb
ab => ab
ab-1 => aB
ba-1 => bA
a-1b-1 => AB
ab2 => abb
ab-2 => aBB
a-1b2 => Abb
a-1b-2 => ABB
ab3 => abbb
a-1b3 => Abbb
Order = 18
Is Group : True
Is Abelian : True
() | a b A B bb BB bbb ab aB bA AB abb aBB Abb ABB abbb Abbb
-------------------------------------------------------------------------------------------
a | A ab () aB abb aBB abbb bA AB b B Abb ABB bb BB Abbb bbb
b | ab bb bA () bbb B BB abb a Abb A abbb aB Abbb AB aBB ABB
A | () bA a AB Abb ABB Abbb b B ab aB bb BB abb aBB bbb abbb
B | aB () AB BB b bbb bb a aBB A ABB ab abbb bA Abbb abb Abb
bb | abb bbb Abb b BB () B abbb ab Abbb bA aBB a ABB A aB AB
BB | aBB B ABB bbb () bb b aB abbb AB Abbb a abb A Abb ab bA
bbb | abbb BB Abbb bb B b () aBB abb ABB Abb aB ab AB bA a A
ab | bA abb b a abbb aB aBB Abb A bb () Abbb AB bbb B ABB BB
aB | AB a B aBB ab abbb abb A ABB () BB bA Abbb b bbb Abb bb
bA | b Abb ab A Abbb AB ABB bb () abb a bbb B abbb aB BB aBB
AB | B A aB ABB bA Abbb Abb () BB a aBB b bbb ab abbb bb abb
abb | Abb abbb bb ab aBB a aB Abbb bA bbb b ABB A BB () AB B
aBB | ABB aB BB abbb a abb ab AB Abbb B bbb A Abb () bb bA b
Abb | bb Abbb abb bA ABB A AB bbb b abbb ab BB () aBB a B aB
ABB | BB AB aBB Abbb A Abb bA B bbb aB abbb () bb a abb b ab
abbb | Abbb aBB bbb abb aB ab a ABB Abb BB bb AB bA B b A ()
Abbb | bbb ABB abbb Abb AB bA A BB bb aBB abb B b aB ab () a
Repr : ()
() => ()
a3 => aaa
a-3 => AAA
b6 => bbbbbb
b-6 => BBBBBB
aba-1b-1 => abAB
ab-1a-1b => aBAb
bab-1a-1 => baBA
ba-1b-1a => bABa
a-1bab-1 => AbaB
a-1b-1ab => ABab
b-1aba-1 => BabA
b-1a-1ba => BAba
Repr : a
a => a
a-2 => AA
bab-1 => baB
b-1ab => Bab
Repr : b
b => b
b-5 => BBBBB
aba-1 => abA
a-1ba => Aba
Repr : A
a-1 => A
a2 => aa
ba-1b-1 => bAB
b-1a-1b => BAb
Repr : B
b-1 => B
b5 => bbbbb
ab-1a-1 => aBA
a-1b-1a => ABa
Repr : bb
b2 => bb
b-4 => BBBB
ab2a-1 => abbA
a-1b2a => Abba
Repr : BB
b-2 => BB
b4 => bbbb
ab-2a-1 => aBBA
a-1b-2a => ABBa
Repr : bbb
b3 => bbb
b-3 => BBB
Repr : ab
ab => ab
ba => ba
Repr : aB
ab-1 => aB
b-1a => Ba
Repr : bA
ba-1 => bA
a-1b => Ab
Repr : AB
a-1b-1 => AB
b-1a-1 => BA
Repr : abb
ab2 => abb
b2a => bba
a-1b2a-1 => AbbA
Repr : aBB
ab-2 => aBB
b-2a => BBa
a-1b-2a-1 => ABBA
Repr : Abb
a-1b2 => Abb
b2a-1 => bbA
ab2a => abba
Repr : ABB
a-1b-2 => ABB
b-2a-1 => BBA
ab-2a => aBBa
ba-1b3 => bAbbb
Repr : abbb
ab3 => abbb
b3a => bbba
Repr : Abbb
a-1b3 => Abbb
b3a-1 => bbbA
ba-1b2 => bAbb
Total Words : 65
Total Time : 828 ms; Total Created Words : 44555
and
G = { (), a, b, c, ab, ac, bc, ca, cb, abc, aca, acb, bca, bcb, cab, abca, abcb, acab, bcab, cabc, abcab, acabc, bcabc, abcabc }
() => ()
a => a
b => b
c => c
ab => ab
ac => ac
bc => bc
ca => ca
cb => cb
abc => abc
aca => aca
acb => acb
bca => bca
bcb => bcb
cab => cab
abca => abca
abcb => abcb
acab => acab
bcab => bcab
cabc => cabc
abcab => abcab
acabc => acabc
bcabc => bcabc
abcabc => abcabc
Order = 24
Is Group : True
Is Abelian : False
() | a b c ab ac bc ca cb abc aca acb bca bcb cab abca abcb acab bcab cabc abcab acabc bcabc abcabc
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------
a | () ab ac b c abc aca acb bc ca cb abca abcb acab bca bcb cab abcab acabc bcab cabc abcabc bcabc
b | ab () bc a abc c bca bcb ac abca abcb ca cb bcab aca acb abcab cab bcabc acab abcabc cabc acabc
c | ca cb () cab aca bcb a b cabc ac acab bcab bc ab bcabc acabc acb bca abc abcabc abcb abca abcab
ab | b a abc () bc ac abca abcb c bca bcb aca acb abcab ca cb bcab acab abcabc cab bcabc acabc cabc
ac | aca acb a acab ca abcb () ab acabc c cab abcab abc b abcabc cabc cb abca bc bcabc bcb bca bcab
bc | bca bcb b bcab abca cb ab () bcabc abc abcab cab c a cabc abcabc abcb ca ac acabc acb aca acab
ca | c cab aca cb () cabc ac acab bcb a b bcabc acabc acb bcab bc ab abcabc abcb bca abc abcab abca
cb | cab c bcb ca cabc () bcab bc aca bcabc acabc a b bca ac acab abcabc ab abca acb abcab abc abcb
abc | abca abcb ab abcab bca acb b a abcabc bc bcab acab ac () acabc bcabc bcb aca c cabc cb ca cab
aca | ac acab ca acb a acabc c cab abcb () ab abcabc cabc cb abcab abc b bcabc bcb abca bc bcab bca
acb | acab ac abcb aca acabc a abcab abc ca abcabc cabc () ab abca c cab bcabc b bca cb bcab bc bcb
bca | bc bcab abca bcb b bcabc abc abcab cb ab () cabc abcabc abcb cab c a acabc acb ca ac acab aca
bcb | bcab bc cb bca bcabc b cab c abca cabc abcabc ab () ca abc abcab acabc a aca abcb acab ac acb
cab | cb ca cabc c bcb aca bcabc acabc () bcab bc ac acab abcabc a b bca acb abcab ab abca abcb abc
abca | abc abcab bca abcb ab abcabc bc bcab acb b a acabc bcabc bcb acab ac () cabc cb aca c cab ca
abcb | abcab abc acb abca abcabc ab acab ac bca acabc bcabc b a aca bc bcab cabc () ca bcb cab c cb
acab | acb aca acabc ac abcb ca abcabc cabc a abcab abc c cab bcabc () ab abca cb bcab b bca bcb bc
bcab | bcb bca bcabc bc cb abca cabc abcabc b cab c abc abcab acabc ab () ca abcb acab a aca acb ac
cabc | bcabc acabc cab abcabc bcab acab cb ca abcab bcb bca acb aca c abcb abca bc ac () abc b a ab
abcab | abcb abca abcabc abc acb bca acabc bcabc ab acab ac bc bcab cabc b a aca bcb cab () ca cb c
acabc | abcabc cabc acab bcabc abcab cab acb aca bcab abcb abca cb ca ac bcb bca abc c a bc ab () b
bcabc | cabc abcabc bcab acabc cab abcab bcb bca acab cb ca abcb abca bc acb aca c abc b ac () ab a
abcabc | acabc bcabc abcab cabc acab bcab abcb abca cab acb aca bcb bca abc cb ca ac bc ab c a b ()
Repr : ()
() => ()
a2 => aa
b2 => bb
c2 => cc
a-2 => AA
b-2 => BB
c-2 => CC
abab => abab
baba => baba
acacac => acacac
bcbcbc => bcbcbc
cacaca => cacaca
cbcbcb => cbcbcb
Repr : a
a => a
a-1 => A
bab => bab
cacac => cacac
Repr : b
b => b
b-1 => B
aba => aba
cbcbc => cbcbc
Repr : c
c => c
c-1 => C
acaca => acaca
bcbcb => bcbcb
Repr : ab
ab => ab
ba => ba
Repr : ac
ac => ac
caca => caca
Repr : bc
bc => bc
cbcb => cbcb
Repr : ca
ca => ca
acac => acac
Repr : cb
cb => cb
bcbc => bcbc
Repr : abc
abc => abc
Repr : aca
aca => aca
cac => cac
Repr : acb
acb => acb
Repr : bca
bca => bca
Repr : bcb
bcb => bcb
cbc => cbc
Repr : cab
cab => cab
Repr : abca
abca => abca
Repr : abcb
abcb => abcb
Repr : acab
acab => acab
Repr : bcab
bcab => bcab
Repr : cabc
cabc => cabc
acabcb => acabcb
bcabca => bcabca
Repr : abcab
abcab => abcab
Repr : acabc
acabc => acabc
cabcb => cabcb
Repr : bcabc
bcabc => bcabc
cabca => cabca
Repr : abcabc
abcabc => abcabc
Total Words : 56
Total Time : 2079 ms; Total Created Words : 98126
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