#19552 ⟨a, b | aab=b, abbaa=aa⟩

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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.

1aba2bab2ba2b2ab3b2a2b3ab4b3a2b4a
11aba2bab2ba2b2ab3b2a2b3ab4b3a2b4a
aaa2b3b2a2b3ab4b3a2b4aba2bab2ba2b2a
bbbab2ba2b2ab3b2a2b3ab4b3a2b4aba2ba
a2a2b2a2ba2bab2ba2b2ab3b2a2b3ab4b3a2b4a
bababa2b4b3a2b4aba2bab2ba2b2ab3b2a2b3a
b2b2b2ab3b2a2b3ab4b3a2b4aba2bab2ba2b2a
ba2ba2b3a2b2ba2b2ab3b2a2b3ab4b3a2b4aba2ba
b2ab2ab2a2ba2bab2ba2b2ab3b2a2b3ab4b3a2b4a
b3b3b3ab4b3a2b4aba2bab2ba2b2ab3b2a2b3a
b2a2b2a2a2b3b2a2b3ab4b3a2b4aba2bab2ba2b2a
b3ab3ab3a2b2ba2b2ab3b2a2b3ab4b3a2b4aba2ba
b4b4b4aba2bab2ba2b2ab3b2a2b3ab4b3a2b4a
b3a2b3a2ba2b4b3a2b4aba2bab2ba2b2ab3b2a2b3a
b4ab4aa2b3b2a2b3ab4b3a2b4aba2bab2ba2b2a

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. b5 ⇒ b [7]
2. ab ⇒ b3 [5]
3. b4a2 ⇒ a2 [6]
4. a3 ⇒ b2a2 [3]
# ab:aab=b,abbaa=aa b/a - -
bbbbb=b
ab=bbb
bbbbaa=aa
aaa=bbaa

Same cardinality

30 unique, 314 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
105218⟨a, b | aab=bb, bbba=a⟩Noc 146
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
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.

2 total

Σ#PresentationMapping
1119584⟨a, b | aab=b, babaa=aa⟩φ(a) = a, φ(b) = b
1119600⟨a, b | aab=b, bbaaa=aa⟩φ(a) = a, φ(b) = b