#1682 ⟨a, b | aba=bb, bab=a⟩

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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.

1ababb2ab2b3ab3b4ab4b5ab5b6b7
11ababb2ab2b3ab3b4ab4b5ab5b6b7
aab3abb4ab2b5ab3b6ab4b7ab5b2aab
bbab5b2ab3abb4ab2b5ab3b6ab4b7b2
ababb2ab2b3ab3b4ab4b5ab5b6ab7abab2
b2b2ab4b3ab5b4ab5abb6ab2b7ab3b2b3
ab2ab2b7ab3b2ab4b3ab5b4ab5abb6ab2ab3
b3b3ab3b4ab4b5ab5b6ab7abb2ab2b3b4
ab3ab3b6ab4b7ab5b2ab3abb4ab2b5ab3ab4
b4b4ab2b5ab3b6ab4b7ab5b2ab3abb4b5
ab4ab4b5ab5b6ab7abb2ab2b3ab3b4ab4ab5
b5b5abb6ab2b7ab3b2ab4b3ab5b4ab5b6
ab5ab5b4ab5abb6ab2b7ab3b2ab4b3ab5a
b6b6ab7abb2ab2b3ab3b4ab4b5ab5b6b7
b7b7ab5b2ab3abb4ab2b5ab3b6ab4b7b2

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. b8 ⇒ b2 [5]
2. ab6 ⇒ a [7]
3. ba ⇒ ab5 [6]
4. a2 ⇒ b3 [3]
# ab:aba=bb,bab=a b/a - -
bbbbbbbb=bb
abbbbbb=a
ba=abbbbb
aa=bbb

Same cardinality

30 unique, 315 total

Σ#PresentationPropertiesDescriptionφ
81123⟨a, b | aa=1, abbbb=b⟩Noc 1459
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)
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
1119734⟨a, b | aba=b, abaab=aa⟩φ(a) = b, φ(b) = a