#7039 ⟨a, b | bb=aa, aaaba=b⟩

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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.

1ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
11ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
aab2abb3ab2b4ab3b5ab4b6ab5b7ab6b8ab7bab8b2
bbab5b2ab6b3ab7b4ab8b5abb6ab2b7ab3b8ab4bab5
ababb7ab2b8ab3bab4b2ab5b3ab6b4ab7b5ab8b6abb7
b2b2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8babb2ab2
ab2ab2b4ab3b5ab4b6ab5b7ab6b8ab7bab8b2abb3ab2b4
b3b3ab7b4ab8b5abb6ab2b7ab3b8ab4bab5b2ab6b3ab7
ab3ab3bab4b2ab5b3ab6b4ab7b5ab8b6abb7ab2b8ab3b
b4b4ab4b5ab5b6ab6b7ab7b8ab8babb2ab2b3ab3b4ab4
ab4ab4b6ab5b7ab6b8ab7bab8b2abb3ab2b4ab3b5ab4b6
b5b5abb6ab2b7ab3b8ab4bab5b2ab6b3ab7b4ab8b5ab
ab5ab5b3ab6b4ab7b5ab8b6abb7ab2b8ab3bab4b2ab5b3
b6b6ab6b7ab7b8ab8babb2ab2b3ab3b4ab4b5ab5b6ab6
ab6ab6b8ab7bab8b2abb3ab2b4ab3b5ab4b6ab5b7ab6b8
b7b7ab3b8ab4bab5b2ab6b3ab7b4ab8b5abb6ab2b7ab3
ab7ab7b5ab8b6abb7ab2b8ab3bab4b2ab5b3ab6b4ab7b5
b8b8ab8babb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ab8
ab8ab8b2abb3ab2b4ab3b5ab4b6ab5b7ab6b8ab7bab8b2

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 [5]
3. a2 ⇒ b2 [1]
# ab:bb=aa,aaaba=b b/a - -
bbbbbbbbb=b
ba=abbbbb
aa=bb

Same cardinality

17 unique, 102 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
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

Isomorphic instances

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

2 total

Σ#PresentationMapping
107044⟨a, b | bb=aa, aabab=a⟩φ(a) = b, φ(b) = a
107053⟨a, b | bb=aa, abbba=b⟩φ(a) = a, φ(b) = b