#16545 ⟨a, b | aba=bb, abb=aaa⟩

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Quick links

  1. Properties
  2. Elements
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality

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.

Cayley table

Idempotents are shown in bold.

1aba2abbaa3a2bababa2baba4a3baba2(ab)2ba3ba2ba5aba3aba2b
11aba2abbaa3a2bababa2baba4a3baba2(ab)2ba3ba2ba5aba3aba2b
aaa2aba3a2babaa4a3ba3aba2(ab)2a5a4a4a3baba3aba2ba4a5a4
bbbaababa2bababa2ba3ba2b(ab)2aba3aba2ba4a5a3baba3a5a4a5a4a5
a2a2a3a2ba4a3ba3a5a4a4a4a3ba4a5a5a4a5a4a5a4a5
abababaa3aba2(ab)2a4aba3aba2ba3ba5a4a5a4a4a5a4a5a4a5a4
bababa2babba3ba2b(ab)2a4a5ba3a3baba3a5a4a4a5a4a5a4a5a4
a3a3a4a3ba5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
a2ba2ba3a4a4a3ba5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
abaabaaba2(ab)2aba3aba2ba3ba5a4aba3a4a5a4a5a5a4a5a4a5a4a5
ba2ba2ba3ba2ba4a5ba3a5a4a4a4a5a4a5a5a4a5a4a5a4a5
babbab(ab)2ba3a3baba3a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
a4a4a5a4a4a5a5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
a3ba3ba4a5a5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
aba2aba2aba3aba2ba5a4aba3a4a5a5a5a4a5a4a4a5a4a5a4a5a4
(ab)2(ab)2a3baba3a4a5a5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
ba3ba3a4a5a5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
ba2bba2bba3a4a4a5a5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
a5a5a4a5a5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
aba3aba3a5a4a4a5a5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
aba2baba2baba3a5a5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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
#RuleProof
1. a6 ⇒ a4 [5]
2. a4b ⇒ a4 [4]
3. a2ba ⇒ a3 [2]
4. ba4 ⇒ a4 [7]
5. b2 ⇒ aba [1]
6. ba3b ⇒ a5 [8]
7. (ba)2 ⇒ (ab)2 [3]
# ab:aba=bb,abb=aaa reversed:a/b
aaaaaa=aaaa
aaaab=aaaa
aaba=aaa
baaaa=aaaa
bb=aba
baaab=aaaaa
baba=abab

Same cardinality

20 unique, 65 total

Σ#PresentationDescriptionRelated
8526⟨a, b | aab=a, bbbb=1⟩Finite non-commutative monoid with 20 elements10 iso, 9 anti-iso
92207⟨a, b | ab=aa, bbbb=b⟩Finite non-commutative monoid with 20 elements1 anti-iso
104095⟨a, b | baa=abb, abab=1⟩Finite non-Abelian group with 20 elements5 iso
104212⟨a, b | aaaaa=1, abbbb=1⟩Isomorphic to ℤ2016 iso
105349⟨a, b | aaa=bb, bab=aa⟩Finite non-commutative monoid with 20 elements1 iso
106728⟨a, b | aba=a, aaaa=bb⟩Finite non-commutative monoid with 20 elements
106764⟨a, b | aba=b, aaaa=bb⟩Finite non-commutative monoid with 20 elements
107117⟨a, b | ab=aa, bbbb=bb⟩Finite non-commutative monoid with 20 elements
1112159⟨a, b | aaaa=aa, abbb=b⟩Finite non-commutative monoid with 20 elements
1112322⟨a, b | aaab=bb, bbba=a⟩Finite non-commutative monoid with 20 elements
1114367⟨a, b | aaaa=a, bbbbb=a⟩Isomorphic to ℕ(20 = 5)
1114407⟨a, b | aaaa=b, bbbbb=a⟩Isomorphic to ℕ(20 = 1)
1114408⟨a, b | aaaa=b, bbbbb=b⟩Isomorphic to ℕ(20 = 4)
1115933⟨a, b | aba=bb, baaab=a⟩Finite non-commutative monoid with 20 elements
1116124⟨a, b | aab=aa, baba=bb⟩Finite non-commutative monoid with 20 elements
1116339⟨a, b | aba=aa, aaaa=bb⟩Finite non-commutative monoid with 20 elements
1116343⟨a, b | aba=aa, aaab=bb⟩Finite non-commutative monoid with 20 elements
1118619⟨a, b | aba=b, aaaaabb=1⟩Finite non-Abelian group with 20 elements3 iso
1119503⟨a, b | aab=a, bbbbb=bb⟩Finite non-commutative monoid with 20 elements
1121047⟨a, b | ab=aa, bbbb=bbb⟩Finite non-commutative monoid with 20 elements