#5300 ⟨a, b | aaa=aa, aba=bb⟩

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Properties

Elements

Elements in the center commute with all other elements.
The zero element z satisfies zy = yz = z for all y.
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. a3 ⇒ a2
  2. b2 ⇒ aba
  3. (ab)2 ⇒ (ba)2
  4. baba2b ⇒ (a2b)2a
  5. ab(a2b)2 ⇒ b(a2b)2a 
# ab:aaa=aa,aba=bb a/b
aaa=aa
bb=aba
abab=baba
babaab=aabaaba
abaabaab=baabaaba

Right Cayley graph

Left Cayley graph

Others with same cardinality

6 unique, 93 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
91695⟨a, b | bab=aa, bbb=b⟩Finite non-commutative monoid with 27 elements1 iso
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