#16020 ⟨a, b | aaa=ab, baab=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.

1aba2baa3ba2a4ba3a5ba4a6ba5ba6
11aba2baa3ba2a4ba3a5ba4a6ba5ba6
aaa2a3a3a4a4a5a5a6a6a5a5a6a5
bbbaba4ba2ba5ba3ba6ba4ba5ba5ba6ba6ba5ba6
a2a2a3a4a4a5a5a6a6a5a5a6a6a5a6
bababa2ba3ba3ba4ba4ba5ba5ba6ba6ba5ba5ba6ba5
a3a3a4a5a5a6a6a5a5a6a6a5a5a6a5
ba2ba2ba3ba4ba4ba5ba5ba6ba6ba5ba5ba6ba6ba5ba6
a4a4a5a6a6a5a5a6a6a5a5a6a6a5a6
ba3ba3ba4ba5ba5ba6ba6ba5ba5ba6ba6ba5ba5ba6ba5
a5a5a6a5a5a6a6a5a5a6a6a5a5a6a5
ba4ba4ba5ba6ba6ba5ba5ba6ba6ba5ba5ba6ba6ba5ba6
a6a6a5a6a6a5a5a6a6a5a5a6a6a5a6
ba5ba5ba6ba5ba5ba6ba6ba5ba5ba6ba6ba5ba5ba6ba5
ba6ba6ba5ba6ba6ba5ba5ba6ba6ba5ba5ba6ba6ba5ba6

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 ⇒ a5 [3]
2. ab ⇒ a3 [1]
3. b2 ⇒ ba4 [2]
# ab:aaa=ab,baab=bb a/b - -
aaaaaaa=aaaaa
ab=aaa
bb=baaaa

Same cardinality

30 unique, 315 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
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.

1 total

Σ#PresentationMapping
1116024⟨a, b | aaa=ab, baba=bb⟩φ(a) = a, φ(b) = b