#1523 ⟨a, b | aab=ba, bbbb=1⟩

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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. a16 ⇒ a
  2. ab ⇒ ba8
  3. b4 ⇒ 1 
# ab:aab=ba,bbbb=1 a/b
aaaaaaaaaaaaaaaa=a
ab=baaaaaaaa
bbbb=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

2 unique, 9 total

Σ#PresentationDescriptionRelated
105442⟨a, b | aaaa=1, ababbb=1⟩Finite non-Abelian group with 64 elements2 iso
1010273⟨a, b | aa=1, ababa=bbb⟩Finite non-commutative monoid with 64 elements5 iso

Other isomorphic instances

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

6 total

Σ#PresentationMapping
103947⟨a, b | abab=ba, bbbb=1⟩φ(a) = bbba, φ(b) = b
1110939⟨a, b | aabb=bab, bbbb=1⟩φ(a) = a, φ(b) = b
1111079⟨a, b | baab=abb, bbbb=1⟩φ(a) = a, φ(b) = bbb
1117101⟨a, b | aaaa=1, aaabba=b⟩φ(a) = b, φ(b) = a
1117626⟨a, b | aaaa=1, aaaba=bb⟩φ(a) = bbb, φ(b) = a
1117648⟨a, b | aaaa=1, abaab=ba⟩φ(a) = bbb, φ(b) = bba