#9083 ⟨a, b | ab=a, bbaaa=bb⟩

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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.

1aba2bab2a3ba2b2aba3b2a2
11aba2bab2a3ba2b2aba3b2a2
aaa2aa3a2aaa3a2aa3
bbbab2ba2b2ab2ba3b2a2b2ab2b2a2
a2a2a3a2aa3a2a2aa3a2a
bababa2baba3ba2bababa3ba2baba3
b2b2b2ab2b2a2b2ab2b2b2a2b2ab2b2a2
a3a3aa3a2aa3a3a2aa3a2
ba2ba2ba3ba2baba3ba2ba2baba3ba2ba
b2ab2ab2a2b2ab2b2a2b2ab2ab2b2a2b2ab2
ba3ba3baba3ba2baba3ba3ba2baba3ba2
b2a2b2a2b2b2a2b2ab2b2a2b2a2b2ab2b2a2b2a

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. ab ⇒ a [1]
2. b3 ⇒ b2 [4]
3. a4 ⇒ a [3]
4. b2a3 ⇒ b2 [2]
# ab:ab=a,bbaaa=bb ab - -
ab=a
bbb=bb
aaaa=a
bbaaa=bb

Same cardinality

29 unique, 476 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
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
1120885⟨a, b | bb=aa, aaaaa=ab⟩Noc 1115
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.

4 total

Σ#PresentationMapping
1125059⟨a, b | ab=a, bbaaab=bb⟩φ(a) = a, φ(b) = b
1125063⟨a, b | ab=a, bbaaba=bb⟩φ(a) = a, φ(b) = b
1125071⟨a, b | ab=a, bbabaa=bb⟩φ(a) = a, φ(b) = b
1125087⟨a, b | ab=a, bbbaaa=bb⟩φ(a) = a, φ(b) = b