#5349 ⟨a, b | aaa=bb, bab=aa⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances

Properties

Finite structure

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.

1ababbab2abaab2babb3ab3bab2b4ab4bab3b5ab5bab4b6bab5
11ababbab2abaab2babb3ab3bab2b4ab4bab3b5ab5bab4b6bab5
aabababbab2abaab2ab5bab3b2ab3bab4b3ab4bab5b4ab5bab2b5ab2b6
bbbab2babab2b3b2bab2ab3b4bab3ab4b5bab4ab5b6bab5ab2b3ab3
abababaab2b2bab3ab3ab2b3bab4ab4b4bab5ab5b5bab2ab2b6bab3ab3bab4
babaab3babab4b2bab2bab5ab5b3bab3ab2b4bab4ab3b5bab5ab4b6bab2b3
b2b2ab2b3ab3bab2b4b3ab4bab3b5ab5bab4b6ab2bab5b3ab3bab2b4bab3
abaababab4b2bab5ab2b3b6bab2ab3b4bab3ab4b5bab4ab5b6bab5ab2b3ab3
ab2ab2bab3ab3bab4b3ab4ab3bab5b4ab5bab2b5ab2bab3b6ab3bab4b3ab4b4
babbabb2bab2b3ab5bab3bab2b4ab2bab4b5ab3bab5b6ab4bab2b3ab5bab3ab2
b3b3bab2b4bab3ab4b5b4bab4ab5b6bab5ab2b3bab2ab3b4bab3ab4b5ab5
ab3ab3b3ab4b4bab5ab5ab4b5bab2ab2b6bab3ab3b3bab4ab4b4bab5ab5bab2
bab2bab2ab5bab3ab2b4bab4bab3ab3b5bab5ab4b6bab2ab5b3bab3ab2b4bab4b5
b4b4ab4b5ab5bab4b6b5ab2bab5b3ab3bab2b4ab4bab3b5ab5bab4b6bab5
ab4ab4bab5ab5bab2b5ab2ab5bab3b6ab3bab4b3ab4bab5b4ab5bab2b5ab2b6
bab3bab3b4bab4b5ab3bab5bab4b6ab4bab2b3ab5bab3b4ab2bab4b5ab3bab5ab4
b5b5bab4b6bab5ab2b3b6bab2ab3b4bab3ab4b5bab4ab5b6bab5ab2b3ab3
ab5ab5b5ab2b6bab3ab3ab2b3bab4ab4b4bab5ab5b5bab2ab2b6bab3ab3bab4
bab4bab4ab3bab5ab4b6bab2bab5ab5b3bab3ab2b4bab4ab3b5bab5ab4b6bab2b3
b6b6ab2b3ab3bab2b4b3ab4bab3b5ab5bab4b6ab2bab5b3ab3bab2b4bab3
bab5bab5b6bab2b3ab5bab3bab2b4ab2bab4b5ab3bab5b6ab4bab2b3ab5bab3ab2

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
Word:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. b7 ⇒ b3 [8]
2. ab6 ⇒ ab2 [7]
3. b2a ⇒ ab2 [6]
4. a2 ⇒ bab [2]
5. (ab)2 ⇒ b2 [4]
6. (ba)2 ⇒ b2 [3]
# ab:aaa=bb,bab=aa b/a - -
bbbbbbb=bbb
abbbbbb=abb
bba=abb
aa=bab
abab=bb
baba=bb

Same cardinality

20 unique, 64 total

Σ#PresentationPropertiesDescriptionφ
8526⟨a, b | aab=a, bbbb=1⟩Noc 2019
92207⟨a, b | ab=aa, bbbb=b⟩Noc 201
104095⟨a, b | baa=abb, abab=1⟩Grp 205
104212⟨a, b | aaaaa=1, abbbb=1⟩Grp Com 20ℤ2016
106728⟨a, b | aba=a, aaaa=bb⟩Noc 20
106764⟨a, b | aba=b, aaaa=bb⟩Noc 20
107117⟨a, b | ab=aa, bbbb=bb⟩Noc 20
1112159⟨a, b | aaaa=aa, abbb=b⟩Noc 20
1112322⟨a, b | aaab=bb, bbba=a⟩Noc 20
1114367⟨a, b | aaaa=a, bbbbb=a⟩Noc Com 20ℕ(20 = 5)
1114407⟨a, b | aaaa=b, bbbbb=a⟩Noc Com 20ℕ(20 = 1)
1114408⟨a, b | aaaa=b, bbbbb=b⟩Noc Com 20ℕ(20 = 4)
1115933⟨a, b | aba=bb, baaab=a⟩Noc 20
1116124⟨a, b | aab=aa, baba=bb⟩Noc 20
1116339⟨a, b | aba=aa, aaaa=bb⟩Noc 20
1116343⟨a, b | aba=aa, aaab=bb⟩Noc 20
1116545⟨a, b | aba=bb, abb=aaa⟩Noc 20
1118619⟨a, b | aba=b, aaaaabb=1⟩Grp 203
1119503⟨a, b | aab=a, bbbbb=bb⟩Noc 20
1121047⟨a, b | ab=aa, bbbb=bbb⟩Noc 20

Isomorphic instances

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

1 total

Σ#PresentationMapping
1116452⟨a, b | aba=bb, abab=aa⟩φ(a) = b, φ(b) = a