#15797 ⟨a, b | aab=bb, bbbba=a⟩

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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

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.

1aba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba8b
11aba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba8b
aaa2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7baa8bab
bba3a2ba4a3ba5a4ba6a5ba7a6ba8a7baa8ba2aba2b
a2a2a3a2ba4a3ba5a4ba6a5ba7a6ba8a7baa8ba2aba2b
ababa4a3ba5a4ba6a5ba7a6ba8a7baa8ba2aba3a2ba3b
a3a3a4a3ba5a4ba6a5ba7a6ba8a7baa8ba2aba3a2ba3b
a2ba2ba5a4ba6a5ba7a6ba8a7baa8ba2aba3a2ba4a3ba4b
a4a4a5a4ba6a5ba7a6ba8a7baa8ba2aba3a2ba4a3ba4b
a3ba3ba6a5ba7a6ba8a7baa8ba2aba3a2ba4a3ba5a4ba5b
a5a5a6a5ba7a6ba8a7baa8ba2aba3a2ba4a3ba5a4ba5b
a4ba4ba7a6ba8a7baa8ba2aba3a2ba4a3ba5a4ba6a5ba6b
a6a6a7a6ba8a7baa8ba2aba3a2ba4a3ba5a4ba6a5ba6b
a5ba5ba8a7baa8ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
a7a7a8a7baa8ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
a6ba6baa8ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba8b
a8a8aa8ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba8b
a7ba7ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7baa8bab
a8ba8ba3a2ba4a3ba5a4ba6a5ba7a6ba8a7baa8ba2aba2b

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. a9 ⇒ a [10]
2. ba ⇒ a3 [9]
3. b2 ⇒ a2b [1]
# ab:aab=bb,bbbba=a a/b - -
aaaaaaaaa=a
ba=aaa
bb=aab

Same cardinality

17 unique, 104 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
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
1120914⟨a, b | bb=aa, ababa=aa⟩Noc 181