#3771 ⟨a, b | aaaa=bb, abab=1⟩

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Properties

Elements

Elements in the center commute with all other elements.
The order of x is the least n such that xn = 1.

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. a24 ⇒ 1
  2. ab ⇒ ba19
  3. b2 ⇒ a4 
# ab:aaaa=bb,abab=1 a/b
aaaaaaaaaaaaaaaaaaaaaaaa=1
ab=baaaaaaaaaaaaaaaaaaa
bb=aaaa

Right Cayley graph

Left Cayley graph

Others with 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

Other 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