#4212 ⟨a, b | aaaaa=1, abbbb=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

1bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19
11bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19
bbb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b191
b2b2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b191b
b3b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b191bb2
b4b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b191bb2b3
b5b5b6b7b8b9b10b11b12b13b14b15b16b17b18b191bb2b3b4
b6b6b7b8b9b10b11b12b13b14b15b16b17b18b191bb2b3b4b5
b7b7b8b9b10b11b12b13b14b15b16b17b18b191bb2b3b4b5b6
b8b8b9b10b11b12b13b14b15b16b17b18b191bb2b3b4b5b6b7
b9b9b10b11b12b13b14b15b16b17b18b191bb2b3b4b5b6b7b8
b10b10b11b12b13b14b15b16b17b18b191bb2b3b4b5b6b7b8b9
b11b11b12b13b14b15b16b17b18b191bb2b3b4b5b6b7b8b9b10
b12b12b13b14b15b16b17b18b191bb2b3b4b5b6b7b8b9b10b11
b13b13b14b15b16b17b18b191bb2b3b4b5b6b7b8b9b10b11b12
b14b14b15b16b17b18b191bb2b3b4b5b6b7b8b9b10b11b12b13
b15b15b16b17b18b191bb2b3b4b5b6b7b8b9b10b11b12b13b14
b16b16b17b18b191bb2b3b4b5b6b7b8b9b10b11b12b13b14b15
b17b17b18b191bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16
b18b18b191bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17
b19b191bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18

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. b20 ⇒ 1 [7]
2. a ⇒ b16 [6]
# ab:aaaaa=1,abbbb=1 b/a - -
bbbbbbbbbbbbbbbbbbbb=1
a=bbbbbbbbbbbbbbbb

Same cardinality

20 unique, 49 total

Σ#PresentationPropertiesDescriptionφ
8526⟨a, b | aab=a, bbbb=1⟩Noc 2019
92207⟨a, b | ab=aa, bbbb=b⟩Noc 201
104095⟨a, b | baa=abb, abab=1⟩Grp 205
105349⟨a, b | aaa=bb, bab=aa⟩Noc 201
106728⟨a, b | aba=a, aaaa=bb⟩Noc 20
106764⟨a, b | aba=b, aaaa=bb⟩Noc 20
107117⟨a, b | ab=aa, bbbb=bb⟩Noc 20
1112159⟨a, b | aaaa=aa, abbb=b⟩Noc 20
1112322⟨a, b | aaab=bb, bbba=a⟩Noc 20
1114367⟨a, b | aaaa=a, bbbbb=a⟩Noc Com 20ℕ(20 = 5)
1114407⟨a, b | aaaa=b, bbbbb=a⟩Noc Com 20ℕ(20 = 1)
1114408⟨a, b | aaaa=b, bbbbb=b⟩Noc Com 20ℕ(20 = 4)
1115933⟨a, b | aba=bb, baaab=a⟩Noc 20
1116124⟨a, b | aab=aa, baba=bb⟩Noc 20
1116339⟨a, b | aba=aa, aaaa=bb⟩Noc 20
1116343⟨a, b | aba=aa, aaab=bb⟩Noc 20
1116545⟨a, b | aba=bb, abb=aaa⟩Noc 20
1118619⟨a, b | aba=b, aaaaabb=1⟩Grp 203
1119503⟨a, b | aab=a, bbbbb=bb⟩Noc 20
1121047⟨a, b | ab=aa, bbbb=bbb⟩Noc 20

Isomorphic instances

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

16 total

Σ#PresentationMapping
104216⟨a, b | aaaaa=1, babbb=1⟩φ(a) = bbbbbbbbbbbbbbbb, φ(b) = b
104217⟨a, b | aaaaa=1, bbabb=1⟩φ(a) = bbbbbbbbbbbbbbbb, φ(b) = b
104381⟨a, b | aaaa=b, bbbbb=1⟩φ(a) = b, φ(b) = bbbb
105447⟨a, b | aaaa=1, abbbbb=1⟩φ(a) = bbbbbbbbbbbbbbb, φ(b) = b
105454⟨a, b | aaaa=1, babbbb=1⟩φ(a) = bbbbbbbbbbbbbbb, φ(b) = b
105456⟨a, b | aaaa=1, bbabbb=1⟩φ(a) = bbbbbbbbbbbbbbb, φ(b) = b
105732⟨a, b | aaaa=1, bbbbb=a⟩φ(a) = bbbbb, φ(b) = b
1111691⟨a, b | aaaa=bb, bbbbb=1⟩φ(a) = b, φ(b) = bbbbbbbbbbbb
1113287⟨a, b | aaaaa=1, aabbbb=1⟩φ(a) = bbbbbbbb, φ(b) = b
1113297⟨a, b | aaaaa=1, abbbba=1⟩φ(a) = bbbbbbbb, φ(b) = b
1113302⟨a, b | aaaaa=1, baabbb=1⟩φ(a) = bbbbbbbb, φ(b) = b
1113306⟨a, b | aaaaa=1, bbaabb=1⟩φ(a) = bbbbbbbb, φ(b) = b
1116606⟨a, b | aaaa=1, abbabbb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbbbbbbbbbbbb
1116608⟨a, b | aaaa=1, abbbabb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbbbbbbbbbbbb
1116623⟨a, b | aaaa=1, babbabb=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbbbbbbbbbbbb
1116624⟨a, b | aaaa=1, babbbab=1⟩φ(a) = bbbbb, φ(b) = bbbbbbbbbbbbbbbbbb