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

Right Cayley graph

Left Cayley graph

Others with 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

Other 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