#3773 ⟨a, b | aaaa=bb, abbb=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

1aa2a3a4a5a6a7a8a9a10a11a12a13
11aa2a3a4a5a6a7a8a9a10a11a12a13
aaa2a3a4a5a6a7a8a9a10a11a12a131
a2a2a3a4a5a6a7a8a9a10a11a12a131a
a3a3a4a5a6a7a8a9a10a11a12a131aa2
a4a4a5a6a7a8a9a10a11a12a131aa2a3
a5a5a6a7a8a9a10a11a12a131aa2a3a4
a6a6a7a8a9a10a11a12a131aa2a3a4a5
a7a7a8a9a10a11a12a131aa2a3a4a5a6
a8a8a9a10a11a12a131aa2a3a4a5a6a7
a9a9a10a11a12a131aa2a3a4a5a6a7a8
a10a10a11a12a131aa2a3a4a5a6a7a8a9
a11a11a12a131aa2a3a4a5a6a7a8a9a10
a12a12a131aa2a3a4a5a6a7a8a9a10a11
a13a131aa2a3a4a5a6a7a8a9a10a11a12

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. a14 ⇒ 1 [4]
2. b ⇒ a9 [3]
# ab:aaaa=bb,abbb=1 a/b - -
aaaaaaaaaaaaaa=1
b=aaaaaaaaa

Same cardinality

30 unique, 151 total

Σ#PresentationPropertiesDescriptionφ
81123⟨a, b | aa=1, abbbb=b⟩Noc 1459
91682⟨a, b | aba=bb, bab=a⟩Noc 141
93034⟨a, b | aa=a, bbbb=ab⟩Noc 142
105218⟨a, b | aab=bb, bbba=a⟩Noc 146
105404⟨a, b | aab=bb, aba=aa⟩Noc 141
106718⟨a, b | aab=b, bbba=aa⟩Noc 142
1112183⟨a, b | aaaa=ab, baab=b⟩Noc 143
1112206⟨a, b | aaaa=bb, abbb=a⟩Noc Com 141
1112207⟨a, b | aaaa=bb, abbb=b⟩Noc Com 141
1112441⟨a, b | aabb=aa, baab=b⟩Noc 143
1112499⟨a, b | abab=aa, abba=b⟩Noc 14
1114383⟨a, b | aaaa=b, aabbb=a⟩Noc Com 14ℕ(14 = 1)15
1114384⟨a, b | aaaa=b, aabbb=b⟩Noc Com 14ℕ(14 = 4)5
1115532⟨a, b | aaa=bb, abbbb=b⟩Noc Com 14ℕ(14 = 3)11
1115539⟨a, b | aaa=bb, babbb=a⟩Noc Com 14
1116020⟨a, b | aaa=ab, baab=bb⟩Noc 141
1116079⟨a, b | aaa=bb, bbbb=ab⟩Noc 14
1116293⟨a, b | aab=bb, abab=aa⟩Noc 14
1116470⟨a, b | aba=bb, bbbb=aa⟩Noc 14
1119552⟨a, b | aab=b, abbaa=aa⟩Noc 142
1120844⟨a, b | ab=aa, bbbbb=aa⟩Noc 141
1120846⟨a, b | ab=aa, bbbbb=ba⟩Noc 14
1121023⟨a, b | ab=aa, bbaa=bbb⟩Noc 141
1121040⟨a, b | ab=aa, bbbb=aaa⟩Noc 143
1121044⟨a, b | ab=aa, bbbb=baa⟩Noc 141
1121046⟨a, b | ab=aa, bbbb=bba⟩Noc 14
1121110⟨a, b | bb=aa, abab=aaa⟩Noc 142
1124186⟨a, b | aa=a, bbbbbbb=a⟩Noc Com 14ℕ(14 = 7)
1124331⟨a, b | aa=b, bbbbbbb=b⟩Noc Com 14ℕ(14 = 2)
1125055⟨a, b | ab=a, bbaaaa=bb⟩Noc 14

Isomorphic instances

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

165 total

Σ#PresentationMapping
103775⟨a, b | aaaa=bb, babb=1⟩φ(a) = bbbbbbbbbbb, φ(b) = b
104369⟨a, b | aaaa=b, aabbb=1⟩φ(a) = bbb, φ(b) = bbbbbbbbbbbb
104372⟨a, b | aaaa=b, ababb=1⟩φ(a) = bbb, φ(b) = bbbbbbbbbbbb
104373⟨a, b | aaaa=b, abbab=1⟩φ(a) = bbb, φ(b) = bbbbbbbbbbbb
104374⟨a, b | aaaa=b, abbba=1⟩φ(a) = bbb, φ(b) = bbbbbbbbbbbb
104377⟨a, b | aaaa=b, baabb=1⟩φ(a) = bbb, φ(b) = bbbbbbbbbbbb
104378⟨a, b | aaaa=b, babab=1⟩φ(a) = bbb, φ(b) = bbbbbbbbbbbb
104815⟨a, b | aaa=bb, abbbb=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
104819⟨a, b | aaa=bb, babbb=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
104820⟨a, b | aaa=bb, bbabb=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
105489⟨a, b | aaab=1, abbbbb=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105505⟨a, b | aaab=1, babbbb=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105513⟨a, b | aaab=1, bbabbb=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105517⟨a, b | aaab=1, bbbabb=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105519⟨a, b | aaab=1, bbbbab=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105520⟨a, b | aaab=1, bbbbba=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105553⟨a, b | aaba=1, abbbbb=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105569⟨a, b | aaba=1, babbbb=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105577⟨a, b | aaba=1, bbabbb=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105581⟨a, b | aaba=1, bbbabb=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105583⟨a, b | aaba=1, bbbbab=1⟩φ(a) = bbbbbbbbb, φ(b) = b
105584⟨a, b | aaba=1, bbbbba=1⟩φ(a) = bbbbbbbbb, φ(b) = b
106016⟨a, b | aaa=b, aabbbb=1⟩φ(a) = bbbbb, φ(b) = b
106022⟨a, b | aaa=b, ababbb=1⟩φ(a) = bbbbb, φ(b) = b
106024⟨a, b | aaa=b, abbabb=1⟩φ(a) = bbbbb, φ(b) = b
106025⟨a, b | aaa=b, abbbab=1⟩φ(a) = bbbbb, φ(b) = b
106026⟨a, b | aaa=b, abbbba=1⟩φ(a) = bbbbb, φ(b) = b
106031⟨a, b | aaa=b, baabbb=1⟩φ(a) = bbbbb, φ(b) = b
106032⟨a, b | aaa=b, bababb=1⟩φ(a) = bbbbb, φ(b) = b
106033⟨a, b | aaa=b, babbab=1⟩φ(a) = bbbbb, φ(b) = b
106035⟨a, b | aaa=b, bbaabb=1⟩φ(a) = bbbbb, φ(b) = b
107407⟨a, b | aab=1, bbbbbbb=1⟩φ(a) = b, φ(b) = bbbbbbbbbbbb
107479⟨a, b | aba=1, bbbbbbb=1⟩φ(a) = b, φ(b) = bbbbbbbbbbbb
108439⟨a, b | aa=b, bbbbbbb=1⟩φ(a) = b, φ(b) = bb
109483⟨a, b | aa=1, abbbbbbb=1⟩φ(a) = bbbbbbb, φ(b) = b
109509⟨a, b | aa=1, babbbbbb=1⟩φ(a) = bbbbbbb, φ(b) = b
109516⟨a, b | aa=1, bbabbbbb=1⟩φ(a) = bbbbbbb, φ(b) = b
109518⟨a, b | aa=1, bbbabbbb=1⟩φ(a) = bbbbbbb, φ(b) = b
109798⟨a, b | aa=1, bbbbbbb=a⟩φ(a) = bbbbbbb, φ(b) = b
1110663⟨a, b | aaaa=bbb, aabb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = bbbbbbbb
1110665⟨a, b | aaaa=bbb, abba=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = bbbbbbbb
1110667⟨a, b | aaaa=bbb, baab=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = bbbbbbbb
1111675⟨a, b | aaaa=bb, aaabb=1⟩φ(a) = bbbb, φ(b) = b
1111677⟨a, b | aaaa=bb, aabab=1⟩φ(a) = bbbb, φ(b) = b
1111678⟨a, b | aaaa=bb, aabba=1⟩φ(a) = bbbb, φ(b) = b
1111680⟨a, b | aaaa=bb, abaab=1⟩φ(a) = bbbb, φ(b) = b
1111681⟨a, b | aaaa=bb, ababa=1⟩φ(a) = bbbb, φ(b) = b
1111686⟨a, b | aaaa=bb, baaab=1⟩φ(a) = bbbb, φ(b) = b
1113324⟨a, b | aaaab=1, aabbbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113332⟨a, b | aaaab=1, ababbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113336⟨a, b | aaaab=1, abbabb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113338⟨a, b | aaaab=1, abbbab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113339⟨a, b | aaaab=1, abbbba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113348⟨a, b | aaaab=1, baabbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113352⟨a, b | aaaab=1, bababb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113354⟨a, b | aaaab=1, babbab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113355⟨a, b | aaaab=1, babbba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113360⟨a, b | aaaab=1, bbaabb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113362⟨a, b | aaaab=1, bbabab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113363⟨a, b | aaaab=1, bbabba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113366⟨a, b | aaaab=1, bbbaab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113367⟨a, b | aaaab=1, bbbaba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113369⟨a, b | aaaab=1, bbbbaa=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113388⟨a, b | aaaba=1, aabbbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113396⟨a, b | aaaba=1, ababbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113400⟨a, b | aaaba=1, abbabb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113402⟨a, b | aaaba=1, abbbab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113403⟨a, b | aaaba=1, abbbba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113412⟨a, b | aaaba=1, baabbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113416⟨a, b | aaaba=1, bababb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113418⟨a, b | aaaba=1, babbab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113419⟨a, b | aaaba=1, babbba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113424⟨a, b | aaaba=1, bbaabb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113426⟨a, b | aaaba=1, bbabab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113427⟨a, b | aaaba=1, bbabba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113430⟨a, b | aaaba=1, bbbaab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113431⟨a, b | aaaba=1, bbbaba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113433⟨a, b | aaaba=1, bbbbaa=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113515⟨a, b | aabaa=1, aabbbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113521⟨a, b | aabaa=1, ababbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113523⟨a, b | aabaa=1, abbabb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113524⟨a, b | aabaa=1, abbbab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113525⟨a, b | aabaa=1, abbbba=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113530⟨a, b | aabaa=1, baabbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113531⟨a, b | aabaa=1, bababb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113532⟨a, b | aabaa=1, babbab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1113534⟨a, b | aabaa=1, bbaabb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbb
1114976⟨a, b | aaa=bb, aaaabb=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
1114978⟨a, b | aaa=bb, aaabab=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
1114979⟨a, b | aaa=bb, aaabba=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
1114982⟨a, b | aaa=bb, aababa=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
1114984⟨a, b | aaa=bb, aabbaa=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
1114988⟨a, b | aaa=bb, abaaab=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
1114999⟨a, b | aaa=bb, baaaab=1⟩φ(a) = bbbbbbbbbb, φ(b) = b
1116888⟨a, b | aabb=1, aaaaaaa=1⟩φ(a) = bbbbbb, φ(b) = b
1117016⟨a, b | abba=1, aaaaaaa=1⟩φ(a) = bbbbbb, φ(b) = b
1117087⟨a, b | abba=1, bbbbbbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = bbbbbbbb
1117417⟨a, b | aabb=1, aaaaaa=b⟩φ(a) = bbbbbbbbbbbbb, φ(b) = bbbbbbbb
1117481⟨a, b | abab=1, aaaaaa=b⟩φ(a) = bbbbbbbbbbbbb, φ(b) = bbbbbbbb
1117545⟨a, b | abba=1, aaaaaa=b⟩φ(a) = bbbbbbbbbbbbb, φ(b) = bbbbbbbb
1117614⟨a, b | abba=1, bbbbbb=a⟩φ(a) = bbbbbb, φ(b) = b
1117947⟨a, b | aabb=1, aaaaa=bb⟩φ(a) = bbbbbb, φ(b) = b
1118074⟨a, b | abba=1, aaaaa=bb⟩φ(a) = bbbbbb, φ(b) = b
1118141⟨a, b | abba=1, bbbbb=aa⟩φ(a) = bbbbbbbbbbbbb, φ(b) = bbbbbbbb
1118687⟨a, b | aba=b, bbbbbbb=1⟩φ(a) = bbbbbbb, φ(b) = bbbbbbbb
1120484⟨a, b | bb=aa, aaaaaaa=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120485⟨a, b | bb=aa, aaaaaab=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120486⟨a, b | bb=aa, aaaaaba=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120487⟨a, b | bb=aa, aaaaabb=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120488⟨a, b | bb=aa, aaaabaa=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120489⟨a, b | bb=aa, aaaabab=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120490⟨a, b | bb=aa, aaaabba=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120491⟨a, b | bb=aa, aaaabbb=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120492⟨a, b | bb=aa, aaabaaa=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120493⟨a, b | bb=aa, aaabaab=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120494⟨a, b | bb=aa, aaababa=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120495⟨a, b | bb=aa, aaababb=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120496⟨a, b | bb=aa, aaabbaa=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120497⟨a, b | bb=aa, aaabbab=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120498⟨a, b | bb=aa, aaabbba=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120499⟨a, b | bb=aa, aabaaab=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120500⟨a, b | bb=aa, aabaaba=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120501⟨a, b | bb=aa, aabaabb=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120502⟨a, b | bb=aa, aababaa=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120503⟨a, b | bb=aa, aababab=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120504⟨a, b | bb=aa, aababba=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120505⟨a, b | bb=aa, aabbaab=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120506⟨a, b | bb=aa, aabbaba=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120507⟨a, b | bb=aa, aabbbaa=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120508⟨a, b | bb=aa, aabbbab=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120509⟨a, b | bb=aa, aabbbba=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120510⟨a, b | bb=aa, abaaaab=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120511⟨a, b | bb=aa, abaaaba=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120512⟨a, b | bb=aa, abaabab=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120513⟨a, b | bb=aa, abaabba=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120514⟨a, b | bb=aa, ababaab=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120515⟨a, b | bb=aa, abababa=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120516⟨a, b | bb=aa, ababbba=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120517⟨a, b | bb=aa, abbaaab=1⟩φ(a) = b, φ(b) = bbbbbbbb
1120518⟨a, b | bb=aa, abbabba=1⟩φ(a) = bbbbbbbb, φ(b) = b
1120519⟨a, b | bb=aa, abbbbba=1⟩φ(a) = b, φ(b) = bbbbbbbb
1122580⟨a, b | aab=1, bbbbbb=aa⟩φ(a) = b, φ(b) = bbbbbbbbbbbb
1122717⟨a, b | aba=1, bbbbbb=aa⟩φ(a) = b, φ(b) = bbbbbbbbbbbb
1123709⟨a, b | aa=b, aabbbbbb=1⟩φ(a) = b, φ(b) = bb
1123732⟨a, b | aa=b, ababbbbb=1⟩φ(a) = b, φ(b) = bb
1123742⟨a, b | aa=b, abbabbbb=1⟩φ(a) = b, φ(b) = bb
1123746⟨a, b | aa=b, abbbabbb=1⟩φ(a) = b, φ(b) = bb
1123748⟨a, b | aa=b, abbbbabb=1⟩φ(a) = b, φ(b) = bb
1123749⟨a, b | aa=b, abbbbbab=1⟩φ(a) = b, φ(b) = bb
1123750⟨a, b | aa=b, abbbbbba=1⟩φ(a) = b, φ(b) = bb
1123766⟨a, b | aa=b, baabbbbb=1⟩φ(a) = b, φ(b) = bb
1123772⟨a, b | aa=b, bababbbb=1⟩φ(a) = b, φ(b) = bb
1123774⟨a, b | aa=b, babbabbb=1⟩φ(a) = b, φ(b) = bb
1123775⟨a, b | aa=b, babbbabb=1⟩φ(a) = b, φ(b) = bb
1123776⟨a, b | aa=b, babbbbab=1⟩φ(a) = b, φ(b) = bb
1123781⟨a, b | aa=b, bbaabbbb=1⟩φ(a) = b, φ(b) = bb
1123782⟨a, b | aa=b, bbababbb=1⟩φ(a) = b, φ(b) = bb
1123783⟨a, b | aa=b, bbabbabb=1⟩φ(a) = b, φ(b) = bb
1123785⟨a, b | aa=b, bbbaabbb=1⟩φ(a) = b, φ(b) = bb
1126102⟨a, b | aa=1, abbbabbbb=1⟩φ(a) = bbbbbbb, φ(b) = bbbbbbbb
1126106⟨a, b | aa=1, abbbbabbb=1⟩φ(a) = bbbbbbb, φ(b) = bbbbbbbb
1126160⟨a, b | aa=1, babbbabbb=1⟩φ(a) = bbbbbbb, φ(b) = bbbbbbbb
1126161⟨a, b | aa=1, babbbbabb=1⟩φ(a) = bbbbbbb, φ(b) = bbbbbbbb
1126176⟨a, b | aa=1, bbabbbabb=1⟩φ(a) = bbbbbbb, φ(b) = bbbbbbbb
1126708⟨a, b | aa=1, bbbabbbb=a⟩φ(a) = bbbbbbb, φ(b) = bbbbbbbb