#1751 ⟨a, b | aaab=1, bbbbb=1⟩

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Quick links

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

Properties

Elements

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 to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 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#PresentationDescriptionRelated
91602⟨a, b | aaa=bb, aba=a⟩Finite non-commutative monoid with 15 elements2 iso
91603⟨a, b | aaa=bb, aba=b⟩Finite non-commutative monoid with 15 elements1 iso
91993⟨a, b | aaa=a, abbb=b⟩Finite non-commutative monoid with 15 elements4 iso
105342⟨a, b | aaa=bb, aba=aa⟩Finite non-commutative monoid with 15 elements1 iso
105344⟨a, b | aaa=bb, aba=bb⟩Finite non-commutative monoid with 15 elements2 iso
106276⟨a, b | aaa=a, bbbbb=a⟩Isomorphic to ℕ(15 = 5)
106316⟨a, b | aaa=b, bbbbb=a⟩Isomorphic to ℕ(15 = 1)
106317⟨a, b | aaa=b, bbbbb=b⟩Isomorphic to ℕ(15 = 3)
1112981⟨a, b | abb=aaa, baaa=b⟩Finite non-commutative monoid with 15 elements3 iso
1112985⟨a, b | abb=aaa, baba=b⟩Finite non-commutative monoid with 15 elements7 iso
1113123⟨a, b | aab=aaa, aba=bb⟩Finite non-commutative monoid with 15 elements
1113191⟨a, b | abb=aaa, bbb=ab⟩Finite non-commutative monoid with 15 elements1 iso
1113192⟨a, b | abb=aaa, bbb=ba⟩Finite non-commutative monoid with 15 elements
1115439⟨a, b | aaa=aa, bbbbb=a⟩Isomorphic to ℕ(15 = 10)
1115503⟨a, b | aaa=ab, bbbbb=a⟩Isomorphic to ℕ(15 = 6)1 iso
1115543⟨a, b | aaa=bb, bbbbb=a⟩Isomorphic to ℕ(15 = 2)1 iso
1116041⟨a, b | aaa=ab, bbbb=aa⟩Finite non-commutative monoid with 15 elements
1116142⟨a, b | aab=aa, bbbb=ab⟩Finite non-commutative monoid with 15 elements
1119374⟨a, b | aaa=b, bbbbb=ab⟩Isomorphic to ℕ(15 = 4)
1119502⟨a, b | aab=a, bbbbb=ba⟩Finite non-commutative monoid with 15 elements1 anti-iso
1120144⟨a, b | aab=b, bbaa=aaa⟩Finite non-commutative monoid with 15 elements
1120819⟨a, b | ab=aa, bbaaa=bb⟩Finite non-commutative monoid with 15 elements6 iso

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