#14698 ⟨a, b | aabb=a, baaaa=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
  9. Anti-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
aaab6abab7ab2aab3abab4ab2ab5ab3ab6ab4ab7ab5a
bbb7b2b8b3bb4b2b5b3b6b4b7b5b8b6b
ababab7ab2aab3abab4ab2ab5ab3ab6ab4ab7ab5aab6ab
b2b2b8b3bb4b2b5b3b6b4b7b5b8b6bb7b2
ab2ab2aab3abab4ab2ab5ab3ab6ab4ab7ab5aab6abab7ab2
b3b3bb4b2b5b3b6b4b7b5b8b6bb7b2b8b3
ab3ab3abab4ab2ab5ab3ab6ab4ab7ab5aab6abab7ab2aab3
b4b4b2b5b3b6b4b7b5b8b6bb7b2b8b3bb4
ab4ab4ab2ab5ab3ab6ab4ab7ab5aab6abab7ab2aab3abab4
b5b5b3b6b4b7b5b8b6bb7b2b8b3bb4b2b5
ab5ab5ab3ab6ab4ab7ab5aab6abab7ab2aab3abab4ab2ab5
b6b6b4b7b5b8b6bb7b2b8b3bb4b2b5b3b6
ab6ab6ab4ab7ab5aab6abab7ab2aab3abab4ab2ab5ab3ab6
b7b7b5b8b6bb7b2b8b3bb4b2b5b3b6b4b7
ab7ab7ab5aab6abab7ab2aab3abab4ab2ab5ab3ab6ab4ab7
b8b8b6bb7b2b8b3bb4b2b5b3b6b4b7b5b8

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 [12]
2. ab8 ⇒ a [13]
3. ba ⇒ b7 [10]
4. a2 ⇒ ab6 [11]
# ab:aabb=a,baaaa=b b/a - -
bbbbbbbbb=b
abbbbbbbb=a
ba=bbbbbbb
aa=abbbbbb

Same cardinality

16 unique, 64 total

Σ#PresentationPropertiesDescriptionφ
104375⟨a, b | aaaa=b, abbbb=1⟩Grp Com 17ℤ1739
106428⟨a, b | aab=b, babbb=a⟩Noc 172
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
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.

1 total

Σ#PresentationMapping
1114762⟨a, b | abab=a, baaaa=b⟩φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

6 total

Σ#PresentationMapping
1114796⟨a, b | abba=a, aaaab=b⟩φ(a) = a, φ(b) = b
1119020⟨a, b | aab=b, abbbba=a⟩φ(a) = abbbbbb, φ(b) = b
1119052⟨a, b | aab=b, babbba=a⟩φ(a) = abbbbbb, φ(b) = b
1119068⟨a, b | aab=b, bbabba=a⟩φ(a) = abbbbbb, φ(b) = b
1119076⟨a, b | aab=b, bbbaba=a⟩φ(a) = abbbbbb, φ(b) = b
1119080⟨a, b | aab=b, bbbbaa=a⟩φ(a) = abbbbbb, φ(b) = b