#5218 ⟨a, b | aab=bb, bbba=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
  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.

1aba2aba3a2ba4a3ba5a4ba6a5ba6b
11aba2aba3a2ba4a3ba5a4ba6a5ba6b
aaa2aba3a2ba4a3ba5a4ba6a5baa6bab
bba3a2ba4a3ba5a4ba6a5baa6ba2aba2b
a2a2a3a2ba4a3ba5a4ba6a5baa6ba2aba2b
ababa4a3ba5a4ba6a5baa6ba2aba3a2ba3b
a3a3a4a3ba5a4ba6a5baa6ba2aba3a2ba3b
a2ba2ba5a4ba6a5baa6ba2aba3a2ba4a3ba4b
a4a4a5a4ba6a5baa6ba2aba3a2ba4a3ba4b
a3ba3ba6a5baa6ba2aba3a2ba4a3ba5a4ba5b
a5a5a6a5baa6ba2aba3a2ba4a3ba5a4ba5b
a4ba4baa6ba2aba3a2ba4a3ba5a4ba6a5ba6b
a6a6aa6ba2aba3a2ba4a3ba5a4ba6a5ba6b
a5ba5ba2aba3a2ba4a3ba5a4ba6a5baa6bab
a6ba6ba3a2ba4a3ba5a4ba6a5baa6ba2aba2b

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. a7 ⇒ a [8]
2. ba ⇒ a3 [7]
3. b2 ⇒ a2b [1]
# ab:aab=bb,bbba=a a/b - -
aaaaaaa=a
ba=aaa
bb=aab

Same cardinality

30 unique, 310 total

Σ#PresentationPropertiesDescriptionφ
81123⟨a, b | aa=1, abbbb=b⟩Noc 1459
91682⟨a, b | aba=bb, bab=a⟩Noc 141
93034⟨a, b | aa=a, bbbb=ab⟩Noc 142
103773⟨a, b | aaaa=bb, abbb=1⟩Grp Com 14ℤ14165
105404⟨a, b | aab=bb, aba=aa⟩Noc 141
106718⟨a, b | aab=b, bbba=aa⟩Noc 142
1112183⟨a, b | aaaa=ab, baab=b⟩Noc 143
1112206⟨a, b | aaaa=bb, abbb=a⟩Noc Com 141
1112207⟨a, b | aaaa=bb, abbb=b⟩Noc Com 141
1112441⟨a, b | aabb=aa, baab=b⟩Noc 143
1112499⟨a, b | abab=aa, abba=b⟩Noc 14
1114383⟨a, b | aaaa=b, aabbb=a⟩Noc Com 14ℕ(14 = 1)15
1114384⟨a, b | aaaa=b, aabbb=b⟩Noc Com 14ℕ(14 = 4)5
1115532⟨a, b | aaa=bb, abbbb=b⟩Noc Com 14ℕ(14 = 3)11
1115539⟨a, b | aaa=bb, babbb=a⟩Noc Com 14
1116020⟨a, b | aaa=ab, baab=bb⟩Noc 141
1116079⟨a, b | aaa=bb, bbbb=ab⟩Noc 14
1116293⟨a, b | aab=bb, abab=aa⟩Noc 14
1116470⟨a, b | aba=bb, bbbb=aa⟩Noc 14
1119552⟨a, b | aab=b, abbaa=aa⟩Noc 142
1120844⟨a, b | ab=aa, bbbbb=aa⟩Noc 141
1120846⟨a, b | ab=aa, bbbbb=ba⟩Noc 14
1121023⟨a, b | ab=aa, bbaa=bbb⟩Noc 141
1121040⟨a, b | ab=aa, bbbb=aaa⟩Noc 143
1121044⟨a, b | ab=aa, bbbb=baa⟩Noc 141
1121046⟨a, b | ab=aa, bbbb=bba⟩Noc 14
1121110⟨a, b | bb=aa, abab=aaa⟩Noc 142
1124186⟨a, b | aa=a, bbbbbbb=a⟩Noc Com 14ℕ(14 = 7)
1124331⟨a, b | aa=b, bbbbbbb=b⟩Noc Com 14ℕ(14 = 2)
1125055⟨a, b | ab=a, bbaaaa=bb⟩Noc 14

Isomorphic instances

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

5 total

Σ#PresentationMapping
1115749⟨a, b | aab=bb, aabba=a⟩φ(a) = a, φ(b) = b
1115761⟨a, b | aab=bb, abbaa=a⟩φ(a) = a, φ(b) = b
1115773⟨a, b | aab=bb, baaba=a⟩φ(a) = a, φ(b) = b
1115777⟨a, b | aab=bb, babaa=a⟩φ(a) = a, φ(b) = b
1115785⟨a, b | aab=bb, bbaaa=a⟩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.

1 total

Σ#PresentationMapping
1112503⟨a, b | abab=aa, baaa=b⟩φ(a) = b, φ(b) = aaa