#16079 ⟨a, b | aaa=bb, bbbb=ab⟩

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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.

1aba2bab2ba2b3b4b5b6b7b8b9
11aba2bab2ba2b3b4b5b6b7b8b9
aaa2b4b2b7b5b3b6b7b8b9b3b4b5
bbbab2ba2b5b3b8b4b5b6b7b8b9b3
a2a2b2b7b5b3b8b6b9b3b4b5b6b7b8
bababa2b5b3b8b6b4b7b8b9b3b4b5b6
b2b2b5b3b8b6b4b9b5b6b7b8b9b3b4
ba2ba2b3b8b6b4b9b7b3b4b5b6b7b8b9
b3b3b6b4b9b7b5b3b6b7b8b9b3b4b5
b4b4b7b5b3b8b6b4b7b8b9b3b4b5b6
b5b5b8b6b4b9b7b5b8b9b3b4b5b6b7
b6b6b9b7b5b3b8b6b9b3b4b5b6b7b8
b7b7b3b8b6b4b9b7b3b4b5b6b7b8b9
b8b8b4b9b7b5b3b8b4b5b6b7b8b9b3
b9b9b5b3b8b6b4b9b5b6b7b8b9b3b4

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. b10 ⇒ b3 [4]
2. ab ⇒ b4 [2]
3. b2a ⇒ b5 [3]
4. a3 ⇒ b2 [1]
# ab:aaa=bb,bbbb=ab b/a - -
bbbbbbbbbb=bbb
ab=bbbb
bba=bbbbb
aaa=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
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