#20885 ⟨a, b | bb=aa, aaaaa=ab⟩

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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.

1abbab2b2ab3b3ab4b4ab5
11abbab2b2ab3b3ab4b4ab5
aab2b4ab3b2ab4b3ab5b4ab3b2a
bbbab2b2ab3b3ab4b4ab5b2ab3
babab3b2ab4b3ab5b4ab3b2ab4b3a
b2b2b2ab3b3ab4b4ab5b2ab3b3ab4
b2ab2ab4b3ab5b4ab3b2ab4b3ab5b4a
b3b3b3ab4b4ab5b2ab3b3ab4b4ab5
b3ab3ab5b4ab3b2ab4b3ab5b4ab3b2a
b4b4b4ab5b2ab3b3ab4b4ab5b2ab3
b4ab4ab3b2ab4b3ab5b4ab3b2ab4b3a
b5b5b2ab3b3ab4b4ab5b2ab3b3ab4

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. b6 ⇒ b3 [4]
2. b5a ⇒ b2a [5]
3. ab ⇒ b4a [2]
4. a2 ⇒ b2 [1]
# ab:bb=aa,aaaaa=ab reversed:b/a - -
bbbbbb=bbb
bbbbba=bba
ab=bbbba
aa=bb

Same cardinality

29 unique, 465 total

Σ#PresentationPropertiesDescriptionφ
91472⟨a, b | aaa=bb, abbb=1⟩Grp Com 11ℤ11259
104155⟨a, b | abb=aaa, bba=b⟩Noc 114
105086⟨a, b | aaa=bb, abbb=a⟩Noc Com 1111
105087⟨a, b | aaa=bb, abbb=b⟩Noc Com 113
105111⟨a, b | aab=aa, baaa=b⟩Noc 117
105322⟨a, b | aaa=ab, abb=bb⟩Noc 113
105335⟨a, b | aaa=ab, bbb=aa⟩Noc 113
105410⟨a, b | aab=bb, bab=aa⟩Noc 112
106292⟨a, b | aaa=b, aabbb=a⟩Noc Com 11ℕ(11 = 1)35
106293⟨a, b | aaa=b, aabbb=b⟩Noc Com 11ℕ(11 = 3)11
106380⟨a, b | aab=a, bbbbb=a⟩Noc Com 11ℕ(11 = 5)1
106834⟨a, b | aba=b, abb=aaa⟩Noc 11
107114⟨a, b | ab=aa, bbbb=aa⟩Noc 111
107116⟨a, b | ab=aa, bbbb=ba⟩Noc 11
108691⟨a, b | aa=b, abbbbb=b⟩Noc Com 11ℕ(11 = 2)32
108777⟨a, b | ab=a, baaaaa=b⟩Noc 1125
109083⟨a, b | ab=a, bbaaa=bb⟩Noc 114
1112179⟨a, b | aaaa=ab, abbb=b⟩Noc Com 11ℕ(11 = 4)20
1115607⟨a, b | aab=aa, bbbbb=a⟩Noc Com 11ℕ(11 = 10)1
1115671⟨a, b | aab=ab, bbbbb=a⟩Noc Com 11ℕ(11 = 6)8
1116148⟨a, b | aab=ab, aaaa=bb⟩Noc 111
1116212⟨a, b | aab=ba, aaaa=bb⟩Noc 11
1118715⟨a, b | aaa=a, aabbba=b⟩Noc 11
1119070⟨a, b | aab=b, bbabbb=a⟩Noc Com 114
1120096⟨a, b | aab=b, abba=aaa⟩Noc 111
1120112⟨a, b | aab=b, baaa=aaa⟩Noc 11
1120168⟨a, b | aab=b, bbbb=aaa⟩Noc Com 11
1125539⟨a, b | ab=a, baaaa=bbb⟩Noc 11
1125635⟨a, b | ab=a, bbbaa=bbb⟩Noc 11

Isomorphic instances

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

7 total

Σ#PresentationMapping
1120889⟨a, b | bb=aa, aaaab=ba⟩φ(a) = b, φ(b) = a
1120893⟨a, b | bb=aa, aaaba=ba⟩φ(a) = b, φ(b) = a
1120896⟨a, b | bb=aa, aaabb=ab⟩φ(a) = a, φ(b) = b
1120903⟨a, b | bb=aa, aabab=ab⟩φ(a) = a, φ(b) = b
1120907⟨a, b | bb=aa, aabba=ab⟩φ(a) = a, φ(b) = b
1120911⟨a, b | bb=aa, abaab=ab⟩φ(a) = a, φ(b) = b
1120915⟨a, b | bb=aa, ababa=ab⟩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.

8 total

Σ#PresentationMapping
1120888⟨a, b | bb=aa, aaaab=ab⟩φ(a) = b, φ(b) = a
1120892⟨a, b | bb=aa, aaaba=ab⟩φ(a) = b, φ(b) = a
1120897⟨a, b | bb=aa, aaabb=ba⟩φ(a) = a, φ(b) = b
1120900⟨a, b | bb=aa, aabaa=ab⟩φ(a) = b, φ(b) = a
1120904⟨a, b | bb=aa, aabab=ba⟩φ(a) = a, φ(b) = b
1120908⟨a, b | bb=aa, aabba=ba⟩φ(a) = a, φ(b) = b
1120912⟨a, b | bb=aa, abaab=ba⟩φ(a) = a, φ(b) = b
1120918⟨a, b | bb=aa, abbba=ab⟩φ(a) = b, φ(b) = a