#6428 ⟨a, b | aab=b, babbb=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
  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.

1ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8
11ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8
aab8abbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7a
bbab5b2ab6b3ab7b4ab5abb6ab2b7ab3b8ab4b
ababb5ab2b6ab3b7ab4b8ab5bab6b2ab7b3ab4ab
b2b2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8ababb2
ab2ab2b2ab3b3ab4b4ab5b5ab6b6ab7b7ab8abbab2
b3b3ab7b4ab5abb6ab2b7ab3b8ab4bab5b2ab6b3
ab3ab3b7ab4b8ab5bab6b2ab7b3ab4abb5ab2b6ab3
b4b4ab4b5ab5b6ab6b7ab7b8ababb2ab2b3ab3b4
ab4ab4b4ab5b5ab6b6ab7b7ab8abbab2b2ab3b3ab4
b5b5abb6ab2b7ab3b8ab4bab5b2ab6b3ab7b4ab5
ab5ab5bab6b2ab7b3ab4abb5ab2b6ab3b7ab4b8ab5
b6b6ab6b7ab7b8ababb2ab2b3ab3b4ab4b5ab5b6
ab6ab6b6ab7b7ab8abbab2b2ab3b3ab4b4ab5b5ab6
b7b7ab3b8ab4bab5b2ab6b3ab7b4ab5abb6ab2b7
ab7ab7b3ab4abb5ab2b6ab3b7ab4b8ab5bab6b2ab7
b8b8ababb2ab2b3ab3b4ab4b5ab5b6ab6b7ab7b8

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 [9]
2. ab8 ⇒ a [8]
3. ba ⇒ ab5 [7]
4. a2 ⇒ b8 [6]
# ab:aab=b,babbb=a b/a - -
bbbbbbbbb=b
abbbbbbbb=a
ba=abbbbb
aa=bbbbbbbb

Same cardinality

16 unique, 69 total

Σ#PresentationPropertiesDescriptionφ
104375⟨a, b | aaaa=b, abbbb=1⟩Grp Com 17ℤ1739
106514⟨a, b | aba=b, baaab=a⟩Noc 17
107152⟨a, b | bb=aa, abab=aa⟩Noc 17
107153⟨a, b | bb=aa, abab=ab⟩Noc 17
107154⟨a, b | bb=aa, abab=ba⟩Noc 17
1113141⟨a, b | aab=aaa, bbb=ab⟩Noc 17
1113142⟨a, b | aab=aaa, bbb=ba⟩Noc 17
1113151⟨a, b | aba=aaa, baa=bb⟩Noc 17
1113156⟨a, b | aba=aaa, bbb=ab⟩Noc 17
1113218⟨a, b | abb=aba, bbb=aa⟩Noc 17
1114395⟨a, b | aaaa=b, abbbb=a⟩Noc Com 17ℕ(17 = 1)2
1114396⟨a, b | aaaa=b, abbbb=b⟩Noc Com 17ℕ(17 = 4)2
1114698⟨a, b | aabb=a, baaaa=b⟩Noc 177
1115488⟨a, b | aaa=ab, babbb=b⟩Noc 173
1116143⟨a, b | aab=aa, bbbb=ba⟩Noc 17
1116521⟨a, b | aab=bb, bab=aaa⟩Noc 17

Isomorphic instances

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

2 total

Σ#PresentationMapping
106440⟨a, b | aab=b, bbbab=a⟩φ(a) = a, φ(b) = b
106520⟨a, b | aba=b, babbb=a⟩φ(a) = abb, φ(b) = b