#6795 ⟨a, b | aba=b, bbbb=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.

1ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
11ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
aab4abb5ab2b6ab3b7ab4b8ab5bab6b2ab7b3ab8b4
bbab5b2ab6b3ab7b4ab8b5abb6ab2b7ab3b8ab4bab5
ababbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abb
b2b2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8babb2ab2
ab2ab2b6ab3b7ab4b8ab5bab6b2ab7b3ab8b4abb5ab2b6
b3b3ab7b4ab8b5abb6ab2b7ab3b8ab4bab5b2ab6b3ab7
ab3ab3b3ab4b4ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3
b4b4ab4b5ab5b6ab6b7ab7b8ab8babb2ab2b3ab3b4ab4
ab4ab4b8ab5bab6b2ab7b3ab8b4abb5ab2b6ab3b7ab4b8
b5b5abb6ab2b7ab3b8ab4bab5b2ab6b3ab7b4ab8b5ab
ab5ab5b5ab6b6ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5
b6b6ab6b7ab7b8ab8babb2ab2b3ab3b4ab4b5ab5b6ab6
ab6ab6b2ab7b3ab8b4abb5ab2b6ab3b7ab4b8ab5bab6b2
b7b7ab3b8ab4bab5b2ab6b3ab7b4ab8b5abb6ab2b7ab3
ab7ab7b7ab8b8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7
b8b8ab8babb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
ab8ab8b4abb5ab2b6ab3b7ab4b8ab5bab6b2ab7b3ab8b4

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. ba ⇒ ab5 [4]
3. a2 ⇒ b4 [2]
# ab:aba=b,bbbb=aa b/a - -
bbbbbbbbb=b
ba=abbbbb
aa=bbbb

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