#20819 ⟨a, b | ab=aa, bbaaa=bb⟩

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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.

1ababbab2ab2babb3ab3bab2b4ab4bab3bab4
11ababbab2ab2babb3ab3bab2b4ab4bab3bab4
aaababab2ab2ab2ab3ab3ab3ab4ab4ab4ab2ab2ab3
bbbab2babb3b3bab2b4b4bab3b2b2bab4b3b4
ababab2ab2ab3ab3ab3ab4ab4ab4ab2ab2ab2ab3ab3ab4
babababbabbab2bab2bab2bab3bab3bab3bab4bab4bab4bab2bab2bab3
b2b2b3b3b4b4b4b2b2b2b3b3b3b4b4b2
ab2ab2ab3ab3ab4ab4ab4ab2ab2ab2ab3ab3ab3ab4ab4ab2
babbabbab2bab2bab3bab3bab3bab4bab4bab4bab2bab2bab2bab3bab3bab4
b3b3b4b4b2b2b2b3b3b3b4b4b4b2b2b3
ab3ab3ab4ab4ab2ab2ab2ab3ab3ab3ab4ab4ab4ab2ab2ab3
bab2bab2bab3bab3bab4bab4bab4bab2bab2bab2bab3bab3bab3bab4bab4bab2
b4b4b2b2b3b3b3b4b4b4b2b2b2b3b3b4
ab4ab4ab2ab2ab3ab3ab3ab4ab4ab4ab2ab2ab2ab3ab3ab4
bab3bab3bab4bab4bab2bab2bab2bab3bab3bab3bab4bab4bab4bab2bab2bab3
bab4bab4bab2bab2bab3bab3bab3bab4bab4bab4bab2bab2bab2bab3bab3bab4

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. a2 ⇒ ab [1]
2. b2a ⇒ b3 [6]
3. aba ⇒ ab2 [3]
4. b5 ⇒ b2 [7]
# ab:ab=aa,bbaaa=bb ba - -
aa=ab
bba=bbb
aba=abb
bbbbb=bb

Same cardinality

22 unique, 184 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
1113191⟨a, b | abb=aaa, bbb=ab⟩Noc 151
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

Isomorphic instances

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

6 total

Σ#PresentationMapping
1120823⟨a, b | ab=aa, bbaab=bb⟩φ(a) = a, φ(b) = b
1120827⟨a, b | ab=aa, bbaba=bb⟩φ(a) = a, φ(b) = b
1120831⟨a, b | ab=aa, bbabb=bb⟩φ(a) = a, φ(b) = b
1120835⟨a, b | ab=aa, bbbaa=bb⟩φ(a) = a, φ(b) = b
1120839⟨a, b | ab=aa, bbbab=bb⟩φ(a) = a, φ(b) = b
1120843⟨a, b | ab=aa, bbbba=bb⟩φ(a) = a, φ(b) = b