#5309 ⟨a, b | aaa=aa, bbb=ab⟩

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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

Properties

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Cayley table

Idempotents are shown in bold.

1aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab6a2
11aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab6a2
aaa2b3a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
bbbab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab5a2
a2a2a2b5a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
bababa2b4ba2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b2b2b2ab3b2a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
ba2ba2ba2b6ba2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b2ab2ab2a2b5b2a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b3b3b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b2a2b2a2b2a2b5b2a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b3ab3ab3a2b6b3a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b4b4b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b3a2b3a2b3a2b6b3a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b4ab4ab4a2b5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b5b5b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b4a2b4a2b4a2b5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b5ab5ab5a2b6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b6b6b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b5a2b5a2b5a2b6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b6ab6ab6a2b5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b6a2b6a2b6a2b5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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. b7 ⇒ b5 [3]
2. ab ⇒ b3 [2]
3. a3 ⇒ a2 [1]
# ab:aaa=aa,bbb=ab b/a
bbbbbbb=bbbbb
ab=bbb
aaa=aa

Same cardinality

8 unique, 64 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
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
1115158⟨a, b | aab=ba, ababab=1⟩Finite non-Abelian group with 21 elements1 iso
1119212⟨a, b | aba=b, baaaab=a⟩Finite non-commutative monoid with 21 elements