#19502 ⟨a, b | aab=a, bbbbb=ba⟩

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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. Anti-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.

1aba2bab2a3ba2b3a4ba3b4a5ba4ba5
11aba2bab2a3ba2b3a4ba3b4a5ba4ba5
aaa2a5a3aa4a4a2a3a5a3a2aa4a5
bbbab2ba2ba5b3ba3bab4ba4ba2baba5ba3ba4
a2a2a3aa4a2a5a5a3a4aa4a3a2a5a
bababa2ba5ba3baba4ba4ba2ba3ba5ba3ba2baba4ba5
b2b2ba5b3baba4b4ba2ba5baba3baba5ba4ba2ba3
a3a3a4a2a5a3aaa4a5a2a5a4a3aa2
ba2ba2ba3baba4ba2ba5ba5ba3ba4baba4ba3ba2ba5ba
b3b3ba4b4ba5ba3bababa4ba5ba2ba5ba4ba3baba2
a4a4a5a3aa4a2a2a5aa3aa5a4a2a3
ba3ba3ba4ba2ba5ba3bababa4ba5ba2ba5ba4ba3baba2
b4b4ba3baba4ba2ba5ba5ba3ba4baba4ba3ba2ba5ba
a5a5aa4a2a5a3a3aa2a4a2aa5a3a4
ba4ba4ba5ba3baba4ba2ba2ba5baba3baba5ba4ba2ba3
ba5ba5baba4ba2ba5ba3ba3baba2ba4ba2baba5ba3ba4

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. a6 ⇒ a [9]
2. ab ⇒ a5 [8]
3. b2a ⇒ ba5 [10]
4. b5 ⇒ ba [2]
# ab:aab=a,bbbbb=ba a/b - -
aaaaaa=a
ab=aaaaa
bba=baaaaa
bbbbb=ba

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
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)
1120144⟨a, b | aab=b, bbaa=aaa⟩Noc 15
1120819⟨a, b | ab=aa, bbaaa=bb⟩Noc 156

Anti-isomorphic instances

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

1 total

Σ#PresentationMapping
1119702⟨a, b | aba=a, bbbbb=ab⟩φ(a) = a, φ(b) = b