#20914 ⟨a, b | bb=aa, ababa=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.

1aba2abbaa3a2babababa4a3b(ab)2(ba)2a5a4bb(ab)2a5b
11aba2abbaa3a2babababa4a3b(ab)2(ba)2a5a4bb(ab)2a5b
aaa2aba3a2babaa4a3ba3b(ab)2a5a4ba5a2a3a5ba2ba3b
bbbaa2a2bbaba3a3ba4(ba)2a3ba4ba5b(ab)2a4ba5ba3a3a4
a2a2a3a2ba4a3ba3ba5a4ba4ba5a3a5ba3a3a4a3ba3ba4b
abababaa3a3b(ab)2a4a4ba5a2a4ba5ba3a2ba5ba3ba4a4a5
babaa2bbaba3ba4(ba)2a4ba5a5b(ab)2a5ba3a5ba2ba3ba4a4a5
a3a3a4a3ba5a4ba4ba3a5ba5ba3a4a3ba4a4a5a4ba4ba5b
a2ba2ba3ba4a4ba5a5a5ba3a3a5ba3ba4a3ba3ba4ba5a5a3
abaabaa3b(ab)2a4ba5a2a5ba3a3a2ba3ba4a3ba3ba4ba5a5a3
babbab(ba)2a3ba5b(ab)2a4ba3a5ba2ba3a4a3ba4a4a5a4ba4ba5b
a4a4a5a4ba3a5ba5ba4a3ba3ba4a5a4ba5a5a3a5ba5ba3b
a3ba3ba4ba5a5ba3a3a3ba4a4a3ba4ba5a4ba4ba5ba3a3a4
(ab)2(ab)2a2a4ba3a2ba5ba4a3ba3ba4a5a4ba5a5a3a5ba5ba3b
(ba)2(ba)2a5b(ab)2a3a5ba2ba4a3ba3ba4a5a4ba5a5a3a5ba5ba3b
a5a5a3a5ba4a3ba3ba5a4ba4ba5a3a5ba3a3a4a3ba3ba4b
a4ba4ba5ba3a3ba4a4a4ba5a5a4ba5ba3a5ba5ba3ba4a4a5
b(ab)2b(ab)2a2ba3a3ba4a4a4ba5a5a4ba5ba3a5ba5ba3ba4a4a5
a5ba5ba3ba4a4ba5a5a5ba3a3a5ba3ba4a3ba3ba4ba5a5a3

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. b2 ⇒ a2 [1]
2. ba2 ⇒ a2b [3]
3. a2ba ⇒ a3b [4]
4. a(ba)2 ⇒ a2 [2]
5. a6 ⇒ a3 [5]
# ab:bb=aa,ababa=aa ab - -
bb=aa
baa=aab
aaba=aaab
ababa=aa
aaaaaa=aaa

Same cardinality

17 unique, 103 total

Σ#PresentationPropertiesDescriptionφ
81121⟨a, b | aa=1, abbba=b⟩Noc 1817
93387⟨a, b | aa=1, abbbbb=b⟩Noc 1831
105521⟨a, b | aaab=1, bbbbbb=1⟩Grp Com 18ℤ1833
106732⟨a, b | aba=a, aaab=bb⟩Noc 18
106788⟨a, b | aba=b, baab=aa⟩Noc 18
106795⟨a, b | aba=b, bbbb=aa⟩Noc 18
107039⟨a, b | bb=aa, aaaba=b⟩Noc 182
108910⟨a, b | aa=a, bbbbb=ab⟩Noc 181
1112187⟨a, b | aaaa=ab, babb=b⟩Noc 182
1115797⟨a, b | aab=bb, bbbba=a⟩Noc 18
1116042⟨a, b | aaa=ab, bbbb=ab⟩Noc 18
1116043⟨a, b | aaa=ab, bbbb=ba⟩Noc 18
1116313⟨a, b | aab=bb, baba=aa⟩Noc 18
1118758⟨a, b | aaa=a, bbbbbb=a⟩Noc Com 18ℕ(18 = 6)
1118830⟨a, b | aaa=b, bbbbbb=a⟩Noc Com 18ℕ(18 = 1)
1118831⟨a, b | aaa=b, bbbbbb=b⟩Noc Com 18ℕ(18 = 3)
1119624⟨a, b | aab=b, bbbba=aa⟩Noc 18

Isomorphic instances

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

1 total

Σ#PresentationMapping
1120916⟨a, b | bb=aa, ababa=bb⟩φ(a) = a, φ(b) = b