#1695 ⟨a, b | bab=aa, bbb=b⟩

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Properties

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
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. b3 ⇒ b
  2. a2 ⇒ bab
  3. (ba)2 ⇒ (ab)2
  4. abab2a ⇒ (b2a)2b
  5. ba(b2a)2 ⇒ a(b2a)2b 
# ab:bab=aa,bbb=b b/a
bbb=b
aa=bab
baba=abab
ababba=bbabbab
babbabba=abbabbab

Right Cayley graph

Left Cayley graph

Others with same cardinality

6 unique, 92 total

Σ#PresentationDescriptionRelated
8450⟨a, b | bab=aa, bbb=1⟩Finite non-commutative monoid with 27 elements27 iso
8752⟨a, b | aaa=1, babb=a⟩Finite non-Abelian group with 27 elements58 iso, 1 anti-iso
105300⟨a, b | aaa=aa, aba=bb⟩Finite non-commutative monoid with 27 elements
1115514⟨a, b | aaa=bb, aabaa=b⟩Finite non-commutative monoid with 27 elements
1118739⟨a, b | aaa=a, abbbbb=b⟩Finite non-commutative monoid with 27 elements
1120314⟨a, b | aba=b, bbbb=aaa⟩Finite non-commutative monoid with 27 elements

Other isomorphic instances

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

1 total

Σ#PresentationMapping
1119242⟨a, b | aaa=a, aaaba=bb⟩φ(a) = b, φ(b) = a