#5521 ⟨a, b | aaab=1, bbbbbb=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

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a171
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a171a
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a171aa2
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a171aa2a3
a5a5a6a7a8a9a10a11a12a13a14a15a16a171aa2a3a4
a6a6a7a8a9a10a11a12a13a14a15a16a171aa2a3a4a5
a7a7a8a9a10a11a12a13a14a15a16a171aa2a3a4a5a6
a8a8a9a10a11a12a13a14a15a16a171aa2a3a4a5a6a7
a9a9a10a11a12a13a14a15a16a171aa2a3a4a5a6a7a8
a10a10a11a12a13a14a15a16a171aa2a3a4a5a6a7a8a9
a11a11a12a13a14a15a16a171aa2a3a4a5a6a7a8a9a10
a12a12a13a14a15a16a171aa2a3a4a5a6a7a8a9a10a11
a13a13a14a15a16a171aa2a3a4a5a6a7a8a9a10a11a12
a14a14a15a16a171aa2a3a4a5a6a7a8a9a10a11a12a13
a15a15a16a171aa2a3a4a5a6a7a8a9a10a11a12a13a14
a16a16a171aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
a17a171aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16

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. a18 ⇒ 1 [9]
2. b ⇒ a15 [8]
# ab:aaab=1,bbbbbb=1 a/b - -
aaaaaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaaaaa

Same cardinality

17 unique, 71 total

Σ#PresentationPropertiesDescriptionφ
81121⟨a, b | aa=1, abbba=b⟩Noc 1817
93387⟨a, b | aa=1, abbbbb=b⟩Noc 1831
106732⟨a, b | aba=a, aaab=bb⟩Noc 18
106788⟨a, b | aba=b, baab=aa⟩Noc 18
106795⟨a, b | aba=b, bbbb=aa⟩Noc 18
107039⟨a, b | bb=aa, aaaba=b⟩Noc 182
108910⟨a, b | aa=a, bbbbb=ab⟩Noc 181
1112187⟨a, b | aaaa=ab, babb=b⟩Noc 182
1115797⟨a, b | aab=bb, bbbba=a⟩Noc 18
1116042⟨a, b | aaa=ab, bbbb=ab⟩Noc 18
1116043⟨a, b | aaa=ab, bbbb=ba⟩Noc 18
1116313⟨a, b | aab=bb, baba=aa⟩Noc 18
1118758⟨a, b | aaa=a, bbbbbb=a⟩Noc Com 18ℕ(18 = 6)
1118830⟨a, b | aaa=b, bbbbbb=a⟩Noc Com 18ℕ(18 = 1)
1118831⟨a, b | aaa=b, bbbbbb=b⟩Noc Com 18ℕ(18 = 3)
1119624⟨a, b | aab=b, bbbba=aa⟩Noc 18
1120914⟨a, b | bb=aa, ababa=aa⟩Noc 181

Isomorphic instances

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

33 total

Σ#PresentationMapping
105585⟨a, b | aaba=1, bbbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaaaaa
106037⟨a, b | aaa=b, bbbbbb=1⟩φ(a) = a, φ(b) = aaa
107259⟨a, b | aaa=1, abbbbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
107273⟨a, b | aaa=1, babbbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
107277⟨a, b | aaa=1, bbabbbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
107278⟨a, b | aaa=1, bbbabbb=1⟩φ(a) = aaaaaaaaaaaa, φ(b) = a
107550⟨a, b | aaa=1, bbbbbb=a⟩φ(a) = aaaaaa, φ(b) = a
1111685⟨a, b | aaaa=bb, abbbb=1⟩φ(a) = aaaaaaaaaaaaaa, φ(b) = a
1111689⟨a, b | aaaa=bb, babbb=1⟩φ(a) = aaaaaaaaaaaaaa, φ(b) = a
1111690⟨a, b | aaaa=bb, bbabb=1⟩φ(a) = aaaaaaaaaaaaaa, φ(b) = a
1113600⟨a, b | aabab=1, bbbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
1113728⟨a, b | abaab=1, bbbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
1113764⟨a, b | ababa=1, bbbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaa, φ(b) = aaa
1113851⟨a, b | aaaa=b, aabbbb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113857⟨a, b | aaaa=b, ababbb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113859⟨a, b | aaaa=b, abbabb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113860⟨a, b | aaaa=b, abbbab=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113861⟨a, b | aaaa=b, abbbba=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113866⟨a, b | aaaa=b, baabbb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113867⟨a, b | aaaa=b, bababb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113868⟨a, b | aaaa=b, babbab=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1113870⟨a, b | aaaa=b, bbaabb=1⟩φ(a) = aaaaaaa, φ(b) = aaaaaaaaaa
1121177⟨a, b | aaa=1, aabbbbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121210⟨a, b | aaa=1, abbabbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121216⟨a, b | aaa=1, abbbbabb=1⟩φ(a) = aaaaaa, φ(b) = a
1121218⟨a, b | aaa=1, abbbbbba=1⟩φ(a) = aaaaaa, φ(b) = a
1121234⟨a, b | aaa=1, baabbbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121242⟨a, b | aaa=1, babbabbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121244⟨a, b | aaa=1, babbbbab=1⟩φ(a) = aaaaaa, φ(b) = a
1121249⟨a, b | aaa=1, bbaabbbb=1⟩φ(a) = aaaaaa, φ(b) = a
1121251⟨a, b | aaa=1, bbabbabb=1⟩φ(a) = aaaaaa, φ(b) = a
1121253⟨a, b | aaa=1, bbbaabbb=1⟩φ(a) = aaaaaa, φ(b) = a
1122325⟨a, b | aaa=1, bbbbbb=aa⟩φ(a) = aaaaaaaaaaaa, φ(b) = a