#19624 ⟨a, b | aab=b, bbbba=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

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.

1abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8a
11abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8a
aab4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4a
bbbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abba
babab5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5a
b2b2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2a
b2ab2ab6ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6a
b3b3b3ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3a
b3ab3ab7ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7a
b4b4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4a
b4ab4ab8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8a
b5b5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5a
b5ab5abab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abba
b6b6b6ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6a
b6ab6ab2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2a
b7b7b7ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7a
b7ab7ab3ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3a
b8b8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8a
b8ab8ab4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4a

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. b9 ⇒ b [6]
2. ab ⇒ b5 [5]
3. a2 ⇒ b4a [2]
# ab:aab=b,bbbba=aa b/a - -
bbbbbbbbb=b
ab=bbbbb
aa=bbbba

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
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)
1120914⟨a, b | bb=aa, ababa=aa⟩Noc 181