#6935 ⟨a, b | bb=aa, aaabab=1⟩

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Quick links

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

Properties

Elements

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

Cayley table

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9ba8a10ba9a11ba10ba11
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9ba8a10ba9a11ba10ba11
aaa2ba7a3ba8a4ba9a5ba10a6ba11a7ba8baa9ba2a10ba3a11ba41ba5ba6
bbbaa2ba2a3ba3a4ba4a5ba5a6ba6a7ba7a8ba8a9ba9a10ba10a11ba111a
a2a2a3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9ba8a10ba9a11ba101ba11abba
bababa2a9ba3a10ba4a11ba51ba6aba7a2ba8a3ba9a4ba10a5ba11a6ba7a8
a3a3a4ba9a5ba10a6ba11a7ba8baa9ba2a10ba3a11ba41ba5aba6a2ba7ba8
ba2ba2ba3a4ba4a5ba5a6ba6a7ba7a8ba8a9ba9a10ba10a11ba111babaa2a3
a4a4a5ba4a6ba5a7ba6a8ba7a9ba8a10ba9a11ba101ba11aba2baa3ba2ba3
ba3ba3ba4a11ba51ba6aba7a2ba8a3ba9a4ba10a5ba11a6ba7baa8ba2a9a10
a5a5a6ba11a7ba8baa9ba2a10ba3a11ba41ba5aba6a2ba7a3ba8a4ba9ba10
ba4ba4ba5a6ba6a7ba7a8ba8a9ba9a10ba10a11ba111babaa2ba2a3ba3a4a5
a6a6a7ba6a8ba7a9ba8a10ba9a11ba101ba11aba2baa3ba2a4ba3a5ba4ba5
ba5ba5ba6aba7a2ba8a3ba9a4ba10a5ba11a6ba7baa8ba2a9ba3a10ba4a111
a7a7a8baa9ba2a10ba3a11ba41ba5aba6a2ba7a3ba8a4ba9a5ba10a6ba11b
ba6ba6ba7a8ba8a9ba9a10ba10a11ba111babaa2ba2a3ba3a4ba4a5ba5a6a7
a8a8a9ba8a10ba9a11ba101ba11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6ba7
ba7ba7ba8a3ba9a4ba10a5ba11a6ba7baa8ba2a9ba3a10ba4a11ba51ba6aa2
a9a9a10ba3a11ba41ba5aba6a2ba7a3ba8a4ba9a5ba10a6ba11a7ba8baba2
ba8ba8ba9a10ba10a11ba111babaa2ba2a3ba3a4ba4a5ba5a6ba6a7ba7a8a9
a10a10a11ba101ba11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9ba8ba9
ba9ba9ba10a5ba11a6ba7baa8ba2a9ba3a10ba4a11ba51ba6aba7a2ba8a3a4
a11a111ba5aba6a2ba7a3ba8a4ba9a5ba10a6ba11a7ba8baa9ba2a10ba3ba4
ba10ba10ba111babaa2ba2a3ba3a4ba4a5ba5a6ba6a7ba7a8ba8a9ba9a10a11
ba11ba11ba7baa8ba2a9ba3a10ba4a11ba51ba6aba7a2ba8a3ba9a4ba10a5a6

Right Cayley graph

Left 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. a12 ⇒ 1 [7]
2. ab ⇒ ba7 [6]
3. b2 ⇒ a2 [1]
# ab:bb=aa,aaabab=1 a/b
aaaaaaaaaaaa=1
ab=baaaaaaa
bb=aa

Same cardinality

6 unique, 75 total

Σ#PresentationDescriptionRelated
8425⟨a, b | aab=ba, bbb=1⟩Finite non-commutative monoid with 24 elements18 iso, 3 anti-iso
105009⟨a, b | aba=bb, aabbb=1⟩Finite non-Abelian group with 24 elements30 iso
1111515⟨a, b | ababa=b, abbaa=1⟩Finite non-Abelian group with 24 elements10 iso
1113372⟨a, b | aaaab=1, bbbbbb=1⟩Isomorphic to ℤ248 iso
1119638⟨a, b | aba=a, aaaab=bb⟩Finite non-commutative monoid with 24 elements
1120252⟨a, b | aba=b, aaaa=bab⟩Finite non-commutative monoid with 24 elements

Isomorphic instances

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

2 total

Σ#PresentationMapping
106939⟨a, b | bb=aa, aababa=1⟩φ(a) = a, φ(b) = b
106944⟨a, b | bb=aa, abaaab=1⟩φ(a) = a, φ(b) = b