#8777 ⟨a, b | ab=a, baaaaa=b⟩

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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
  9. 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.

1aba2baa3ba2a4ba3a5ba4
11aba2baa3ba2a4ba3a5ba4
aaa2aa3a2a4a3a5a4aa5
bbbabba2baba3ba2ba4ba3bba4
a2a2a3a2a4a3a5a4aa5a2a
bababa2baba3ba2ba4ba3bba4bab
a3a3a4a3a5a4aa5a2aa3a2
ba2ba2ba3ba2ba4ba3bba4babba2ba
a4a4a5a4aa5a2aa3a2a4a3
ba3ba3ba4ba3bba4babba2baba3ba2
a5a5aa5a2aa3a2a4a3a5a4
ba4ba4bba4babba2baba3ba2ba4ba3

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. b2 ⇒ b [4]
3. a6 ⇒ a [3]
4. ba5 ⇒ b [2]
# ab:ab=a,baaaaa=b ab - -
ab=a
bb=b
aaaaaa=a
baaaaa=b

Same cardinality

29 unique, 455 total

Σ#PresentationPropertiesDescriptionφ
91472⟨a, b | aaa=bb, abbb=1⟩Grp Com 11ℤ11259
104155⟨a, b | abb=aaa, bba=b⟩Noc 114
105086⟨a, b | aaa=bb, abbb=a⟩Noc Com 1111
105087⟨a, b | aaa=bb, abbb=b⟩Noc Com 113
105111⟨a, b | aab=aa, baaa=b⟩Noc 117
105322⟨a, b | aaa=ab, abb=bb⟩Noc 113
105335⟨a, b | aaa=ab, bbb=aa⟩Noc 113
105410⟨a, b | aab=bb, bab=aa⟩Noc 112
106292⟨a, b | aaa=b, aabbb=a⟩Noc Com 11ℕ(11 = 1)35
106293⟨a, b | aaa=b, aabbb=b⟩Noc Com 11ℕ(11 = 3)11
106380⟨a, b | aab=a, bbbbb=a⟩Noc Com 11ℕ(11 = 5)1
106834⟨a, b | aba=b, abb=aaa⟩Noc 11
107114⟨a, b | ab=aa, bbbb=aa⟩Noc 111
107116⟨a, b | ab=aa, bbbb=ba⟩Noc 11
108691⟨a, b | aa=b, abbbbb=b⟩Noc Com 11ℕ(11 = 2)32
109083⟨a, b | ab=a, bbaaa=bb⟩Noc 114
1112179⟨a, b | aaaa=ab, abbb=b⟩Noc Com 11ℕ(11 = 4)20
1115607⟨a, b | aab=aa, bbbbb=a⟩Noc Com 11ℕ(11 = 10)1
1115671⟨a, b | aab=ab, bbbbb=a⟩Noc Com 11ℕ(11 = 6)8
1116148⟨a, b | aab=ab, aaaa=bb⟩Noc 111
1116212⟨a, b | aab=ba, aaaa=bb⟩Noc 11
1118715⟨a, b | aaa=a, aabbba=b⟩Noc 11
1119070⟨a, b | aab=b, bbabbb=a⟩Noc Com 114
1120096⟨a, b | aab=b, abba=aaa⟩Noc 111
1120112⟨a, b | aab=b, baaa=aaa⟩Noc 11
1120168⟨a, b | aab=b, bbbb=aaa⟩Noc Com 11
1120885⟨a, b | bb=aa, aaaaa=ab⟩Noc 1115
1125539⟨a, b | ab=a, baaaa=bbb⟩Noc 11
1125635⟨a, b | ab=a, bbbaa=bbb⟩Noc 11

Isomorphic instances

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

20 total

Σ#PresentationMapping
1114456⟨a, b | aaab=a, babbb=b⟩φ(a) = baa, φ(b) = a
1114464⟨a, b | aaab=a, bbabb=b⟩φ(a) = baa, φ(b) = a
1114468⟨a, b | aaab=a, bbbab=b⟩φ(a) = baa, φ(b) = a
1114470⟨a, b | aaab=a, bbbba=b⟩φ(a) = baa, φ(b) = a
1114598⟨a, b | aaba=a, bbbba=b⟩φ(a) = baa, φ(b) = a
1114700⟨a, b | aabb=a, baaab=b⟩φ(a) = a, φ(b) = baa
1114702⟨a, b | aabb=a, baaba=b⟩φ(a) = a, φ(b) = baa
1114706⟨a, b | aabb=a, babaa=b⟩φ(a) = a, φ(b) = baa
1114714⟨a, b | aabb=a, bbaaa=b⟩φ(a) = a, φ(b) = baa
1114764⟨a, b | abab=a, baaab=b⟩φ(a) = a, φ(b) = baa
1114766⟨a, b | abab=a, baaba=b⟩φ(a) = a, φ(b) = baa
1114770⟨a, b | abab=a, babaa=b⟩φ(a) = a, φ(b) = baa
1114778⟨a, b | abab=a, bbaaa=b⟩φ(a) = a, φ(b) = baa
1118897⟨a, b | aab=a, baaaaa=b⟩φ(a) = baaaa, φ(b) = a
1124463⟨a, b | ab=a, baaaaab=b⟩φ(a) = a, φ(b) = b
1124465⟨a, b | ab=a, baaaaba=b⟩φ(a) = a, φ(b) = b
1124469⟨a, b | ab=a, baaabaa=b⟩φ(a) = a, φ(b) = b
1124477⟨a, b | ab=a, baabaaa=b⟩φ(a) = a, φ(b) = b
1124493⟨a, b | ab=a, babaaaa=b⟩φ(a) = a, φ(b) = b
1124525⟨a, b | ab=a, bbaaaaa=b⟩φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

5 total

Σ#PresentationMapping
1114568⟨a, b | aaba=a, abbbb=b⟩φ(a) = baa, φ(b) = a
1114800⟨a, b | abba=a, aaabb=b⟩φ(a) = a, φ(b) = baa
1114804⟨a, b | abba=a, aabab=b⟩φ(a) = a, φ(b) = baa
1114810⟨a, b | abba=a, abaab=b⟩φ(a) = a, φ(b) = baa
1119091⟨a, b | aba=a, aaaaab=b⟩φ(a) = baaaa, φ(b) = a