#18619 ⟨a, b | aba=b, aaaaabb=1⟩

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Contents

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

Properties

Finite structure

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

Cayley table

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9ba8ba9
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9ba8ba9
aaa2ba9a3ba4baa5ba2a6ba3a7ba4a8ba5a9ba61ba7ba8
bbbaa5ba2a6ba3a7ba4a8ba5a9ba61ba7aba8a2ba9a3a4
a2a2a3ba8a4ba9a5ba6baa7ba2a8ba3a9ba41ba5aba6ba7
bababa2a4ba3a5ba4a6ba5a7ba6a8ba7a9ba81ba9aba2a3
a3a3a4ba7a5ba8a6ba9a7ba8baa9ba21ba3aba4a2ba5ba6
ba2ba2ba3a3ba4a4ba5a5ba6a6ba7a7ba8a8ba9a9b1baaa2
a4a4a5ba6a6ba7a7ba8a8ba9a9b1baaba2a2ba3a3ba4ba5
ba3ba3ba4a2ba5a3ba6a4ba7a5ba8a6ba9a7ba8baa9ba21a
a5a5a6ba5a7ba6a8ba7a9ba81ba9aba2baa3ba2a4ba3ba4
ba4ba4ba5aba6a2ba7a3ba8a4ba9a5ba6baa7ba2a8ba3a91
a6a6a7ba4a8ba5a9ba61ba7aba8a2ba9a3ba4baa5ba2ba3
ba5ba5ba61ba7aba8a2ba9a3ba4baa5ba2a6ba3a7ba4a8a9
a7a7a8ba3a9ba41ba5aba6a2ba7a3ba8a4ba9a5ba6baba2
ba6ba6ba7a9ba81ba9aba2baa3ba2a4ba3a5ba4a6ba5a7a8
a8a8a9ba21ba3aba4a2ba5a3ba6a4ba7a5ba8a6ba9a7bba
ba7ba7ba8a8ba9a9b1baaba2a2ba3a3ba4a4ba5a5ba6a6a7
a9a91baaba2a2ba3a3ba4a4ba5a5ba6a6ba7a7ba8a8ba9b
ba8ba8ba9a7ba8baa9ba21ba3aba4a2ba5a3ba6a4ba7a5a6
ba9ba9ba6baa7ba2a8ba3a9ba41ba5aba6a2ba7a3ba8a4a5

Right Cayley graph

Left 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. a10 ⇒ 1 [14]
2. ab ⇒ ba9 [15]
3. b2 ⇒ a5 [13]
# ab:aba=b,aaaaabb=1 a/b - -
aaaaaaaaaa=1
ab=baaaaaaaaa
bb=aaaaa

Same cardinality

20 unique, 62 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
104212⟨a, b | aaaaa=1, abbbb=1⟩Grp Com 20ℤ2016
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
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.

3 total

Σ#PresentationMapping
1118622⟨a, b | aba=b, aaaabba=1⟩φ(a) = a, φ(b) = b
1118628⟨a, b | aba=b, aaabbaa=1⟩φ(a) = a, φ(b) = b
1118668⟨a, b | aba=b, baaaaab=1⟩φ(a) = a, φ(b) = b