#5442 ⟨a, b | aaaa=1, ababbb=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. b16 ⇒ 1 [16]
2. b2a ⇒ ab2 [8]
3. aba ⇒ b13 [15]
4. ba2 ⇒ a2b9 [13]
5. a3 ⇒ bab3 [3]
# ab:aaaa=1,ababbb=1 b/a
bbbbbbbbbbbbbbbb=1
bba=abb
aba=bbbbbbbbbbbbb
baa=aabbbbbbbbb
aaa=babbb

Same cardinality

2 unique, 13 total

Σ#PresentationDescriptionRelated
91523⟨a, b | aab=ba, bbbb=1⟩Finite non-commutative monoid with 64 elements6 iso
1010273⟨a, b | aa=1, ababa=bbb⟩Finite non-commutative monoid with 64 elements5 iso

Isomorphic instances

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

2 total

Σ#PresentationMapping
105445⟨a, b | aaaa=1, abbbab=1⟩φ(a) = a, φ(b) = b
105452⟨a, b | aaaa=1, bababb=1⟩φ(a) = a, φ(b) = b