#3771 ⟨a, b | aaaa=bb, abab=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. a24 ⇒ 1 [13]
2. ab ⇒ ba19 [12]
3. b2 ⇒ a4 [1]
# ab:aaaa=bb,abab=1 a/b
aaaaaaaaaaaaaaaaaaaaaaaa=1
ab=baaaaaaaaaaaaaaaaaaa
bb=aaaa

Same cardinality

4 unique, 13 total

Σ#PresentationDescriptionRelated
1114993⟨a, b | aaa=bb, ababbb=1⟩Finite non-Abelian group with 48 elements2 iso
1116966⟨a, b | abab=1, aaabbba=1⟩Finite non-Abelian group with 48 elements6 iso
1117503⟨a, b | abab=1, aabbaa=b⟩Finite non-Abelian group with 48 elements
1121751⟨a, b | aaa=1, abbbbbb=b⟩Finite non-commutative monoid with 48 elements1 iso

Isomorphic instances

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

5 total

Σ#PresentationMapping
109464⟨a, b | aa=1, ababbbbb=1⟩φ(a) = ba, φ(b) = a
109481⟨a, b | aa=1, abbbbbab=1⟩φ(a) = ba, φ(b) = a
109504⟨a, b | aa=1, bababbbb=1⟩φ(a) = ba, φ(b) = a
109514⟨a, b | aa=1, bbababbb=1⟩φ(a) = ba, φ(b) = a
109786⟨a, b | aa=1, babbbbb=a⟩φ(a) = ba, φ(b) = a