#11058 ⟨a, b | abba=bbb, baba=1⟩

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Quick links

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

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. b15 ⇒ 1 [14]
2. ba ⇒ ab4 [12]
3. a2 ⇒ b10 [15]
# ab:abba=bbb,baba=1 b/a
bbbbbbbbbbbbbbb=1
ba=abbbb
aa=bbbbbbbbbb

Same cardinality

8 unique, 115 total

Σ#PresentationDescriptionRelated
91470⟨a, b | aaa=bb, abab=1⟩Finite non-Abelian group with 30 elements73 iso
91907⟨a, b | aab=a, bbbbb=1⟩Finite non-commutative monoid with 30 elements11 iso, 4 anti-iso
92443⟨a, b | aaa=1, abbbb=b⟩Finite non-commutative monoid with 30 elements14 iso, 5 anti-iso
106560⟨a, b | aaa=a, bbbb=ab⟩Finite non-commutative monoid with 30 elements
107013⟨a, b | ab=aa, bbbbb=b⟩Finite non-commutative monoid with 30 elements
1113115⟨a, b | bbb=aaa, aaba=b⟩Finite non-commutative monoid with 30 elements
1115979⟨a, b | aaa=aa, bbbb=ab⟩Finite non-commutative monoid with 30 elements
1120847⟨a, b | ab=aa, bbbbb=bb⟩Finite non-commutative monoid with 30 elements

Isomorphic instances

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

8 total

Σ#PresentationMapping
1116958⟨a, b | abab=1, aaaabba=1⟩φ(a) = b, φ(b) = a
1116964⟨a, b | abab=1, aaabbaa=1⟩φ(a) = b, φ(b) = a
1116974⟨a, b | abab=1, aabbaaa=1⟩φ(a) = b, φ(b) = a
1116991⟨a, b | abab=1, abbaaaa=1⟩φ(a) = b, φ(b) = a
1116999⟨a, b | abab=1, abbbbba=1⟩φ(a) = a, φ(b) = b
1117012⟨a, b | abab=1, bbaaaaa=1⟩φ(a) = b, φ(b) = a
1118014⟨a, b | abab=1, aaaab=ba⟩φ(a) = b, φ(b) = a
1118057⟨a, b | abab=1, baaaa=ab⟩φ(a) = b, φ(b) = a