#1751 ⟨a, b | aaab=1, bbbbb=1⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality
  9. Isomorphic instances

Properties

Commutative structure

Finite structure

The order of x is the least n such that xn = 1.

Staircase diagram

Cayley table

1aa2a3a4a5a6a7a8a9a10a11a12a13a14
11aa2a3a4a5a6a7a8a9a10a11a12a13a14
aaa2a3a4a5a6a7a8a9a10a11a12a13a141
a2a2a3a4a5a6a7a8a9a10a11a12a13a141a
a3a3a4a5a6a7a8a9a10a11a12a13a141aa2
a4a4a5a6a7a8a9a10a11a12a13a141aa2a3
a5a5a6a7a8a9a10a11a12a13a141aa2a3a4
a6a6a7a8a9a10a11a12a13a141aa2a3a4a5
a7a7a8a9a10a11a12a13a141aa2a3a4a5a6
a8a8a9a10a11a12a13a141aa2a3a4a5a6a7
a9a9a10a11a12a13a141aa2a3a4a5a6a7a8
a10a10a11a12a13a141aa2a3a4a5a6a7a8a9
a11a11a12a13a141aa2a3a4a5a6a7a8a9a10
a12a12a13a141aa2a3a4a5a6a7a8a9a10a11
a13a13a141aa2a3a4a5a6a7a8a9a10a11a12
a14a141aa2a3a4a5a6a7a8a9a10a11a12a13

Right Cayley graph

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a15 ⇒ 1 [8]
2. b ⇒ a12 [7]
# ab:aaab=1,bbbbb=1 a/b - -
aaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaa

Same cardinality

22 unique, 52 total

Σ#PresentationPropertiesDescriptionφ
91602⟨a, b | aaa=bb, aba=a⟩Noc 152
91603⟨a, b | aaa=bb, aba=b⟩Noc 151
91993⟨a, b | aaa=a, abbb=b⟩Noc 154
105342⟨a, b | aaa=bb, aba=aa⟩Noc 151
105344⟨a, b | aaa=bb, aba=bb⟩Noc 152
106276⟨a, b | aaa=a, bbbbb=a⟩Noc Com 15ℕ(15 = 5)
106316⟨a, b | aaa=b, bbbbb=a⟩Noc Com 15ℕ(15 = 1)
106317⟨a, b | aaa=b, bbbbb=b⟩Noc Com 15ℕ(15 = 3)
1112981⟨a, b | abb=aaa, baaa=b⟩Noc 153
1112985⟨a, b | abb=aaa, baba=b⟩Noc 157
1113123⟨a, b | aab=aaa, aba=bb⟩Noc 15
1113191⟨a, b | abb=aaa, bbb=ab⟩Noc 151
1113192⟨a, b | abb=aaa, bbb=ba⟩Noc 15
1115439⟨a, b | aaa=aa, bbbbb=a⟩Noc Com 15ℕ(15 = 10)
1115503⟨a, b | aaa=ab, bbbbb=a⟩Noc Com 15ℕ(15 = 6)1
1115543⟨a, b | aaa=bb, bbbbb=a⟩Noc Com 15ℕ(15 = 2)1
1116041⟨a, b | aaa=ab, bbbb=aa⟩Noc 15
1116142⟨a, b | aab=aa, bbbb=ab⟩Noc 15
1119374⟨a, b | aaa=b, bbbbb=ab⟩Noc Com 15ℕ(15 = 4)
1119502⟨a, b | aab=a, bbbbb=ba⟩Noc 151
1120144⟨a, b | aab=b, bbaa=aaa⟩Noc 15
1120819⟨a, b | ab=aa, bbaaa=bb⟩Noc 156

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

138 total

Σ#PresentationMapping
91783⟨a, b | aaba=1, bbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
91875⟨a, b | aaa=b, bbbbb=1⟩φ(a) = a, φ(b) = aaa
92305⟨a, b | aaa=1, abbbbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
92312⟨a, b | aaa=1, babbbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
92314⟨a, b | aaa=1, bbabbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
92454⟨a, b | aaa=1, bbbbb=a⟩φ(a) = aaaaa, φ(b) = a
104206⟨a, b | aaaaa=1, aabbb=1⟩φ(a) = aaaaaa, φ(b) = a
104209⟨a, b | aaaaa=1, ababb=1⟩φ(a) = aaaaaa, φ(b) = a
104210⟨a, b | aaaaa=1, abbab=1⟩φ(a) = aaaaaa, φ(b) = a
104211⟨a, b | aaaaa=1, abbba=1⟩φ(a) = aaaaaa, φ(b) = a
104214⟨a, b | aaaaa=1, baabb=1⟩φ(a) = aaaaaa, φ(b) = a
104215⟨a, b | aaaaa=1, babab=1⟩φ(a) = aaaaaa, φ(b) = a
104232⟨a, b | aaaab=1, abbbb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104240⟨a, b | aaaab=1, babbb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104244⟨a, b | aaaab=1, bbabb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104246⟨a, b | aaaab=1, bbbab=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104247⟨a, b | aaaab=1, bbbba=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104266⟨a, b | aaaba=1, babbb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104270⟨a, b | aaaba=1, bbabb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104272⟨a, b | aaaba=1, bbbab=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104304⟨a, b | aabaa=1, bbabb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
104821⟨a, b | aaa=bb, bbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaa
107237⟨a, b | aaa=1, aabbbbb=1⟩φ(a) = aaaaa, φ(b) = a
107254⟨a, b | aaa=1, abbabbb=1⟩φ(a) = aaaaa, φ(b) = a
107256⟨a, b | aaa=1, abbbabb=1⟩φ(a) = aaaaa, φ(b) = a
107258⟨a, b | aaa=1, abbbbba=1⟩φ(a) = aaaaa, φ(b) = a
107267⟨a, b | aaa=1, baabbbb=1⟩φ(a) = aaaaa, φ(b) = a
107271⟨a, b | aaa=1, babbabb=1⟩φ(a) = aaaaa, φ(b) = a
107272⟨a, b | aaa=1, babbbab=1⟩φ(a) = aaaaa, φ(b) = a
107275⟨a, b | aaa=1, bbaabbb=1⟩φ(a) = aaaaa, φ(b) = a
107821⟨a, b | aaa=1, bbbbb=aa⟩φ(a) = aaaaaaaaaa, φ(b) = a
1110666⟨a, b | aaaa=bbb, abbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1110668⟨a, b | aaaa=bbb, babb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
1111307⟨a, b | aaabb=b, bbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1111395⟨a, b | aabab=b, bbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1111497⟨a, b | abaab=b, bbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1111529⟨a, b | ababa=b, bbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1111595⟨a, b | baaab=b, bbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1111819⟨a, b | aaab=bb, bbbbb=1⟩φ(a) = a, φ(b) = aaa
1112870⟨a, b | bbb=aaa, aaaaa=1⟩φ(a) = aaaaaa, φ(b) = a
1112871⟨a, b | bbb=aaa, aaaab=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
1112872⟨a, b | bbb=aaa, aaaba=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
1112873⟨a, b | bbb=aaa, aaabb=1⟩φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
1112874⟨a, b | bbb=aaa, aabaa=1⟩φ(a) = aaaaaaaaaaa, φ(b) = a
1112875⟨a, b | bbb=aaa, aabab=1⟩φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
1112876⟨a, b | bbb=aaa, aabba=1⟩φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
1112877⟨a, b | bbb=aaa, abaab=1⟩φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
1112878⟨a, b | bbb=aaa, ababa=1⟩φ(a) = aaaaaaaaaaa, φ(b) = aaaaaa
1112879⟨a, b | bbb=aaa, abbba=1⟩φ(a) = aaaaaa, φ(b) = a
1113280⟨a, b | aaaaa=1, aaabbb=1⟩φ(a) = aaaaaaaaa, φ(b) = a
1113286⟨a, b | aaaaa=1, aabbba=1⟩φ(a) = aaaaaaaaa, φ(b) = a
1113300⟨a, b | aaaaa=1, baaabb=1⟩φ(a) = aaaaaaaaa, φ(b) = a
1113826⟨a, b | aaaa=a, abbbbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1113833⟨a, b | aaaa=a, babbbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1113835⟨a, b | aaaa=a, bbabbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1113844⟨a, b | aaaa=b, aaabbb=1⟩φ(a) = aaaa, φ(b) = a
1113847⟨a, b | aaaa=b, aababb=1⟩φ(a) = aaaa, φ(b) = a
1113849⟨a, b | aaaa=b, aabbab=1⟩φ(a) = aaaa, φ(b) = a
1113850⟨a, b | aaaa=b, aabbba=1⟩φ(a) = aaaa, φ(b) = a
1113854⟨a, b | aaaa=b, abaabb=1⟩φ(a) = aaaa, φ(b) = a
1113855⟨a, b | aaaa=b, ababab=1⟩φ(a) = aaaa, φ(b) = a
1113856⟨a, b | aaaa=b, ababba=1⟩φ(a) = aaaa, φ(b) = a
1113858⟨a, b | aaaa=b, abbaab=1⟩φ(a) = aaaa, φ(b) = a
1113864⟨a, b | aaaa=b, baaabb=1⟩φ(a) = aaaa, φ(b) = a
1113865⟨a, b | aaaa=b, baabab=1⟩φ(a) = aaaa, φ(b) = a
1113968⟨a, b | aaab=b, abbbbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1113984⟨a, b | aaab=b, babbbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1113992⟨a, b | aaab=b, bbabbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1113996⟨a, b | aaab=b, bbbabb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1113998⟨a, b | aaab=b, bbbbab=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1113999⟨a, b | aaab=b, bbbbba=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1114096⟨a, b | aaba=b, abbbbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1114112⟨a, b | aaba=b, babbbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1114120⟨a, b | aaba=b, bbabbb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1114124⟨a, b | aaba=b, bbbabb=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1114126⟨a, b | aaba=b, bbbbab=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1114127⟨a, b | aaba=b, bbbbba=1⟩φ(a) = aaaaaaaaaa, φ(b) = a
1114980⟨a, b | aaa=bb, aaabbb=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1114983⟨a, b | aaa=bb, aababb=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1114985⟨a, b | aaa=bb, aabbab=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1114986⟨a, b | aaa=bb, aabbba=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1114990⟨a, b | aaa=bb, abaabb=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1114992⟨a, b | aaa=bb, ababba=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1114994⟨a, b | aaa=bb, abbaab=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1115000⟨a, b | aaa=bb, baaabb=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1115001⟨a, b | aaa=bb, baabab=1⟩φ(a) = a, φ(b) = aaaaaaaaa
1117222⟨a, b | aaab=1, abbbbb=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117254⟨a, b | aaab=1, babbbb=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117270⟨a, b | aaab=1, bbabbb=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117278⟨a, b | aaab=1, bbbabb=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117282⟨a, b | aaab=1, bbbbab=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117284⟨a, b | aaab=1, bbbbba=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117287⟨a, b | aaab=1, bbbbbb=b⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117350⟨a, b | aaba=1, abbbbb=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117382⟨a, b | aaba=1, babbbb=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117398⟨a, b | aaba=1, bbabbb=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117406⟨a, b | aaba=1, bbbabb=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117410⟨a, b | aaba=1, bbbbab=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117412⟨a, b | aaba=1, bbbbba=a⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1117415⟨a, b | aaba=1, bbbbbb=b⟩φ(a) = a, φ(b) = aaaaaaaaaaaa
1118231⟨a, b | aaa=b, aaabbbb=1⟩φ(a) = a, φ(b) = aaa
1118238⟨a, b | aaa=b, aababbb=1⟩φ(a) = a, φ(b) = aaa
1118241⟨a, b | aaa=b, aabbabb=1⟩φ(a) = a, φ(b) = aaa
1118243⟨a, b | aaa=b, aabbbab=1⟩φ(a) = a, φ(b) = aaa
1118244⟨a, b | aaa=b, aabbbba=1⟩φ(a) = a, φ(b) = aaa
1118251⟨a, b | aaa=b, abaabbb=1⟩φ(a) = a, φ(b) = aaa
1118254⟨a, b | aaa=b, abababb=1⟩φ(a) = a, φ(b) = aaa
1118255⟨a, b | aaa=b, ababbab=1⟩φ(a) = a, φ(b) = aaa
1118256⟨a, b | aaa=b, ababbba=1⟩φ(a) = a, φ(b) = aaa
1118259⟨a, b | aaa=b, abbaabb=1⟩φ(a) = a, φ(b) = aaa
1118260⟨a, b | aaa=b, abbabab=1⟩φ(a) = a, φ(b) = aaa
1118261⟨a, b | aaa=b, abbabba=1⟩φ(a) = a, φ(b) = aaa
1118263⟨a, b | aaa=b, abbbaab=1⟩φ(a) = a, φ(b) = aaa
1118271⟨a, b | aaa=b, baaabbb=1⟩φ(a) = a, φ(b) = aaa
1118273⟨a, b | aaa=b, baababb=1⟩φ(a) = a, φ(b) = aaa
1118274⟨a, b | aaa=b, baabbab=1⟩φ(a) = a, φ(b) = aaa
1118276⟨a, b | aaa=b, babaabb=1⟩φ(a) = a, φ(b) = aaa
1118277⟨a, b | aaa=b, bababab=1⟩φ(a) = a, φ(b) = aaa
1118282⟨a, b | aaa=b, bbaaabb=1⟩φ(a) = a, φ(b) = aaa
1121197⟨a, b | aaa=1, ababbabb=1⟩φ(a) = aaaaa, φ(b) = aaaaaa
1121207⟨a, b | aaa=1, abbababb=1⟩φ(a) = aaaaa, φ(b) = aaaaaa
1121208⟨a, b | aaa=1, abbabbab=1⟩φ(a) = aaaaa, φ(b) = aaaaaa
1121239⟨a, b | aaa=1, bababbab=1⟩φ(a) = aaaaa, φ(b) = aaaaaa
1121706⟨a, b | aaa=1, aabbbbb=a⟩φ(a) = aaaaaaaaaa, φ(b) = a
1121730⟨a, b | aaa=1, ababbbb=a⟩φ(a) = aaaaaaaaaa, φ(b) = a
1121740⟨a, b | aaa=1, abbabbb=a⟩φ(a) = aaaaaaaaaa, φ(b) = a
1121744⟨a, b | aaa=1, abbbabb=a⟩φ(a) = aaaaaaaaaa, φ(b) = a
1121746⟨a, b | aaa=1, abbbbab=a⟩φ(a) = aaaaaaaaaa, φ(b) = a
1121748⟨a, b | aaa=1, abbbbba=a⟩φ(a) = aaaaaaaaaa, φ(b) = a
1121782⟨a, b | aaa=1, bbaabbb=a⟩φ(a) = aaaaaaaaaa, φ(b) = a
1122046⟨a, b | aab=1, bbbbbbb=a⟩φ(a) = aaaaaaa, φ(b) = a
1122190⟨a, b | aba=1, bbbbbbb=a⟩φ(a) = aaaaaaa, φ(b) = a
1122288⟨a, b | aaa=1, abbbbb=aa⟩φ(a) = aaaaa, φ(b) = a
1122321⟨a, b | aaa=1, bbabbb=aa⟩φ(a) = aaaaa, φ(b) = a
1123751⟨a, b | aa=b, abbbbbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a
1123777⟨a, b | aa=b, babbbbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a
1123784⟨a, b | aa=b, bbabbbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a
1123786⟨a, b | aa=b, bbbabbbb=1⟩φ(a) = aaaaaaaa, φ(b) = a