#25055 ⟨a, b | ab=a, bbaaaa=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

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.

1aba2bab2a3ba2b2aa4ba3b2a2ba4b2a3
11aba2bab2a3ba2b2aa4ba3b2a2ba4b2a3
aaa2aa3a2aa4a3a2aa4a3aa4
bbbab2ba2b2ab2ba3b2a2b2aba4b2a3b2a2b2b2a3
a2a2a3a2a4a3a2aa4a3a2aa4a2a
bababa2baba3ba2baba4ba3ba2baba4ba3baba4
b2b2b2ab2b2a2b2ab2b2a3b2a2b2ab2b2a3b2a2b2b2a3
a3a3a4a3aa4a3a2aa4a3a2aa3a2
ba2ba2ba3ba2ba4ba3ba2baba4ba3ba2baba4ba2ba
b2ab2ab2a2b2ab2a3b2a2b2ab2b2a3b2a2b2ab2b2a3b2ab2
a4a4aa4a2aa4a3a2aa4a3a2a4a3
ba3ba3ba4ba3baba4ba3ba2baba4ba3ba2baba3ba2
b2a2b2a2b2a3b2a2b2b2a3b2a2b2ab2b2a3b2a2b2ab2b2a2b2a
ba4ba4baba4ba2baba4ba3ba2baba4ba3ba2ba4ba3
b2a3b2a3b2b2a3b2ab2b2a3b2a2b2ab2b2a3b2a2b2ab2a3b2a2

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. a5 ⇒ a [3]
4. b2a4 ⇒ b2 [2]
# ab:ab=a,bbaaaa=bb ab - -
ab=a
bbb=bb
aaaaa=a
bbaaaa=bb

Same cardinality

30 unique, 316 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
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)