#6101 ⟨a, b | aab=a, bbbbbb=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.
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.

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 ⇒ a [8]
2. ab ⇒ a6 [7]
3. b6 ⇒ 1 [2]
# ab:aab=a,bbbbbb=1 a/b
aaaaaaa=a
ab=aaaaaa
bbbbbb=1

Same cardinality

2 unique, 9 total

Σ#PresentationDescriptionRelated
1114981⟨a, b | aaa=bb, aabaab=1⟩Finite non-Abelian group with 42 elements7 iso
1120647⟨a, b | ab=aa, bbbbbb=b⟩Finite non-commutative monoid with 42 elements

Isomorphic instances

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

3 total

Σ#PresentationMapping
106909⟨a, b | ab=aa, bbbbbb=1⟩φ(a) = bba, φ(b) = b
1114256⟨a, b | abab=a, bbbbbb=1⟩φ(a) = aa, φ(b) = b
1115364⟨a, b | aba=ab, bbbbbb=1⟩φ(a) = ba, φ(b) = b