#9737 ⟨a, b | aa=1, ababbba=b⟩

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Properties

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Complete 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
  1. b7 ⇒ b
  2. bab6 ⇒ ba
  3. b6a ⇒ ab6
  4. a2 ⇒ 1
  5. aba ⇒ bab3
  6. ab3a ⇒ b5ab
  7. ab5a ⇒ b3ab5
  8. bab2a ⇒ ab4ab5
  9. bab4a ⇒ ab2ab3 
# ab:aa=1,ababbba=b b/a
bbbbbbb=b
babbbbbb=ba
bbbbbba=abbbbbb
aa=1
aba=babbb
abbba=bbbbbab
abbbbba=bbbabbbbb
babba=abbbbabbbbb
babbbba=abbabbb

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 2 total

Σ#PresentationDescriptionRelated
1118415⟨a, b | aab=a, bbbbbbb=1⟩Finite non-commutative monoid with 56 elements1 iso

Other isomorphic instances

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

7 total

Σ#PresentationMapping
1010007⟨a, b | aa=1, ababbb=ba⟩φ(a) = a, φ(b) = b
1010019⟨a, b | aa=1, abbbab=ba⟩φ(a) = a, φ(b) = bab
1010326⟨a, b | aa=1, babbb=aba⟩φ(a) = a, φ(b) = b
1126587⟨a, b | aa=1, abababba=b⟩φ(a) = a, φ(b) = abbbbb
1127116⟨a, b | aa=1, abababb=ba⟩φ(a) = a, φ(b) = abbbbb
1127140⟨a, b | aa=1, abbabab=ba⟩φ(a) = a, φ(b) = ab
1127728⟨a, b | aa=1, bababb=aba⟩φ(a) = a, φ(b) = abbbbb