#5342 ⟨a, b | aaa=bb, aba=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.

1aba2abbaa3a2bba2baba4ba3ba2ba5a6
11aba2abbaa3a2bba2baba4ba3ba2ba5a6
aaa2aba3a2ba2a4ba3a3a2ba5a4ba3a6a4
bbbaa3ba2baba4ba3ba2ba5a4a4a6a5a5a6
a2a2a3a2ba4ba3a3a5a4a4ba3a6a5a4a4a5
ababa2a4a3a2ba5a4ba3a6a5a5a4a6a6a4
bababa2babba3ba2bba2a4a6ba3ba2ba5a4a6a6a4
a3a3a4ba3a5a4a4a6a5a5a4a4a6a5a5a6
a2ba2ba3a5a4ba3a6a5a4a4a6a6a5a4a4a5
ba2ba2ba3ba2ba4a6ba3a5a4a4a6a6a5a4a4a5
babbabba2a4ba3ba2ba5a4a6a6a5a5a4a6a6a4
a4a4a5a4a6a5a5a4a6a6a5a5a4a6a6a4
ba3ba3a4a6a5a4a4a6a5a5a4a4a6a5a5a6
ba2bba2bba3a5a4a6a6a5a4a4a6a6a5a4a4a5
a5a5a6a5a4a6a6a5a4a4a6a6a5a4a4a5
a6a6a4a6a5a4a4a6a5a5a4a4a6a5a5a6

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. a7 ⇒ a4 [4]
2. ba4 ⇒ a4 [5]
3. aba ⇒ a2 [2]
4. a3b ⇒ ba3 [3]
5. b2 ⇒ a3 [1]
# ab:aaa=bb,aba=aa a/b - -
aaaaaaa=aaaa
baaaa=aaaa
aba=aa
aaab=baaa
bb=aaa

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
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
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
1116347⟨a, b | aba=aa, aaba=bb⟩φ(a) = a, φ(b) = b