#15158 ⟨a, b | aab=ba, ababab=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

1aba2bab2a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4ba6b2a5b2a6
11aba2bab2a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4ba6b2a5b2a6
aaa2ba4a3ba5b2a2a4ba6b2a3a5bb2a4a6bab2a51ba2b2a6ba3b2b2a
bbbab2ba2b2a1ba3b2a2aba4b2a3a2ba5b2a4a3ba6b2a5a4b2a6a5a6
a2a2a3baa4ba2b2a4a5ba3b2a5a6ba4b2a61ba5b2aba6b2abb2a2b2a3
bababa2b2a4ba3b2a5a2ba4b2a6a3ba5b2a4ba6b2aa5bb2a2a6b2a31a
b2b2b2a1b2a2abb2a3a2bab2a4a3ba2b2a5a4ba3b2a6a5ba4a6ba5ba6
a3a3a4ba5a5ba6b2a6a6bb21bab2aaba2b2a2a2ba3b2a3ba4b2a4b2a5
ba2ba2ba3b2aba4b2a2a4ba5b2a3a5ba6b2a4a6bb2a51bab2a6ab2a2a3
b2ab2ab2a2a4b2a3a5ba2b2a4a6ba3b2a51ba4b2a6aba5b2a2ba6a3bba
a4a4a5ba2a6ba3b2a1ba4b2a2aba5b2a3a2ba6b2a4a3bb2a5bab2a6b2
ba3ba3ba4b2a5ba5b2a6a6ba6b21bb2aabab2a2a2ba2b2a3a3b2a4a4a5
b2a2b2a2b2a3ab2a4a2ba4b2a5a3ba5b2a6a4ba6b2a5bb2aa6ba1ba2ba3
a5a5a6ba61bb2a3abab2a4a2ba2b2a5a3ba3b2a6a4ba4b2ba5b2ab2a2
ba4ba4ba5b2a2ba6b2a3abb2a4a2bab2a5a3ba2b2a6a4ba3b2a5b2aa61
b2a3b2a3b2a4a5b2a5a6ba6b2a61bb2abab2aa2ba2b2a2a3ba3a4ba4ba5
a6a61ba3aba4b2a5a2ba5b2a6a3ba6b2a4bb2aa5bab2a2ba2b2a3b2a4
ba5ba5ba6b2a6bb2a3bab2aa4ba2b2a2a5ba3b2a3a6ba4b2a41b2a5aa2
b2a4b2a4b2a5a2b2a6a3bab2a4ba2b2aa5ba3b2a2a6ba4b2a31ba5aba6b
ba6ba6bb2a3bab2a4a5ba2b2a5a6ba3b2a61ba4b2aba5b2aa2b2a2a3a4
b2a5b2a5b2a6a6b21ba3b2aaba4b2a2a2ba5b2a3a3ba6b2a4a4ba5baba2
b2a6b2a6b2a3b2aa4ba5b2a2a5ba6b2a3a6bb2a41bab2a5aba2a2ba3ba4

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. a7 ⇒ 1 [9]
2. ab ⇒ ba4 [7]
3. b3 ⇒ 1 [4]
# ab:aab=ba,ababab=1 a/b
aaaaaaa=1
ab=baaaa
bbb=1

Same cardinality

8 unique, 63 total

Σ#PresentationDescriptionRelated
8402⟨a, b | aaa=ab, bbb=1⟩Finite non-commutative monoid with 21 elements12 iso, 27 anti-iso
91599⟨a, b | aaa=ab, bbb=b⟩Finite non-commutative monoid with 21 elements1 anti-iso
105309⟨a, b | aaa=aa, bbb=ab⟩Finite non-commutative monoid with 21 elements
106265⟨a, b | aaa=a, abbbb=b⟩Finite non-commutative monoid with 21 elements2 iso
1111127⟨a, b | aaaaa=b, abbbb=1⟩Isomorphic to ℤ2113 iso
1113091⟨a, b | bab=aaa, abba=b⟩Finite non-commutative monoid with 21 elements
1113248⟨a, b | bab=aaa, bbb=aa⟩Finite non-commutative monoid with 21 elements
1119212⟨a, b | aba=b, baaaab=a⟩Finite non-commutative monoid with 21 elements

Isomorphic instances

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

1 total

Σ#PresentationMapping
1115179⟨a, b | aab=ba, bababa=1⟩φ(a) = a, φ(b) = b