#13191 ⟨a, b | abb=aaa, bbb=ab⟩

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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.

1aba2bab2ba2b2ab3b2a2b3ab4b3a2b5b6
11aba2bab2ba2b2ab3b2a2b3ab4b3a2b5b6
aaa2b3b4b3ab4b3a2b6b5b6b5b6b5b5b6
bbbab2ba2b2ab3b2a2b3ab4b3a2b6b5b6b6b5
a2a2b4b5b6b5b6b5b6b5b6b5b6b5b5b6
bababa2b4b5b6b5b6b5b6b5b6b5b6b6b5
b2b2b2ab3b2a2b3ab4b3a2b6b5b6b5b6b5b5b6
ba2ba2b5b6b5b6b5b6b5b6b5b6b5b6b6b5
b2ab2ab2a2b5b6b5b6b5b6b5b6b5b6b5b5b6
b3b3b3ab4b3a2b6b5b6b5b6b5b6b5b6b6b5
b2a2b2a2b6b5b6b5b6b5b6b5b6b5b6b5b5b6
b3ab3ab3a2b6b5b6b5b6b5b6b5b6b5b6b6b5
b4b4b6b5b6b5b6b5b6b5b6b5b6b5b5b6
b3a2b3a2b5b6b5b6b5b6b5b6b5b6b5b6b6b5
b5b5b5b6b5b6b5b6b5b6b5b6b5b6b6b5
b6b6b6b5b6b5b6b5b6b5b6b5b6b5b5b6

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 ⇒ b5 [5]
2. ab ⇒ b3 [2]
3. b4a ⇒ b6 [4]
4. a3 ⇒ b4 [3]
# ab:abb=aaa,bbb=ab b/a - -
bbbbbbb=bbbbb
ab=bbb
bbbba=bbbbbb
aaa=bbbb

Same cardinality

22 unique, 189 total

Σ#PresentationPropertiesDescriptionφ
91602⟨a, b | aaa=bb, aba=a⟩Noc 152
91603⟨a, b | aaa=bb, aba=b⟩Noc 151
91751⟨a, b | aaab=1, bbbbb=1⟩Grp Com 15ℤ15138
91993⟨a, b | aaa=a, abbb=b⟩Noc 154
105342⟨a, b | aaa=bb, aba=aa⟩Noc 151
105344⟨a, b | aaa=bb, aba=bb⟩Noc 152
106276⟨a, b | aaa=a, bbbbb=a⟩Noc Com 15ℕ(15 = 5)
106316⟨a, b | aaa=b, bbbbb=a⟩Noc Com 15ℕ(15 = 1)
106317⟨a, b | aaa=b, bbbbb=b⟩Noc Com 15ℕ(15 = 3)
1112981⟨a, b | abb=aaa, baaa=b⟩Noc 153
1112985⟨a, b | abb=aaa, baba=b⟩Noc 157
1113123⟨a, b | aab=aaa, aba=bb⟩Noc 15
1113192⟨a, b | abb=aaa, bbb=ba⟩Noc 15
1115439⟨a, b | aaa=aa, bbbbb=a⟩Noc Com 15ℕ(15 = 10)
1115503⟨a, b | aaa=ab, bbbbb=a⟩Noc Com 15ℕ(15 = 6)1
1115543⟨a, b | aaa=bb, bbbbb=a⟩Noc Com 15ℕ(15 = 2)1
1116041⟨a, b | aaa=ab, bbbb=aa⟩Noc 15
1116142⟨a, b | aab=aa, bbbb=ab⟩Noc 15
1119374⟨a, b | aaa=b, bbbbb=ab⟩Noc Com 15ℕ(15 = 4)
1119502⟨a, b | aab=a, bbbbb=ba⟩Noc 151
1120144⟨a, b | aab=b, bbaa=aaa⟩Noc 15
1120819⟨a, b | ab=aa, bbaaa=bb⟩Noc 156

Isomorphic instances

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

1 total

Σ#PresentationMapping
1113249⟨a, b | bab=aaa, bbb=ab⟩φ(a) = a, φ(b) = b