#21110 ⟨a, b | bb=aa, abab=aaa⟩

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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. 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.

1aba2abbaa3ababa2baba4ba3(ba)2ba4
11aba2abbaa3ababa2baba4ba3(ba)2ba4
aaa2aba3ba2abaa4ba3ba4a3a4ba4a4ba4
bbbaa2ba2baba3ba3(ba)2a4ba4ba4a4ba4a4
a2a2a3ba2a4ba4ba3a4ba4ba4a4a4ba4a4ba4
abababaa3ba4a3a4ba4a4a4ba4ba4a4ba4a4
bababa2babba3a4(ba)2ba4a4a4ba3ba4a4ba4a4
a3a3a4ba4a4ba4ba4a4ba4ba4a4a4ba4a4ba4
abaababa4a3ba4a4a4ba4a4a4ba4ba4a4ba4a4
ba2ba2ba3a4ba4a4a4ba4a4a4ba4ba4a4ba4a4
babbab(ba)2ba3a4ba3ba4a4ba4ba4a4a4ba4a4ba4
a4a4a4ba4a4ba4ba4a4ba4ba4a4a4ba4a4ba4
ba3ba3ba4a4ba4a4a4ba4a4a4ba4ba4a4ba4a4
(ba)2(ba)2a4ba3a4ba4ba4a4ba4ba4a4a4ba4a4ba4
ba4ba4ba4a4ba4a4a4ba4a4a4ba4ba4a4ba4a4

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. a5 ⇒ a4 [7]
2. aba2 ⇒ ba4 [5]
3. a2b ⇒ ba2 [3]
4. b2 ⇒ a2 [1]
5. (ab)2 ⇒ a3 [2]
# ab:bb=aa,abab=aaa a/b - -
aaaaa=aaaa
abaa=baaaa
aab=baa
bb=aa
abab=aaa

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
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
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

Anti-isomorphic instances

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

2 total

Σ#PresentationMapping
1121111⟨a, b | bb=aa, abab=aab⟩φ(a) = b, φ(b) = a
1121113⟨a, b | bb=aa, abab=baa⟩φ(a) = b, φ(b) = a