#4095 ⟨a, b | baa=abb, abab=1⟩

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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.
The order of x is the least n such that xn = 1.

Cayley table

1aba2abbab2a3a2babaab2b2aa3ba2baa2b2ab2aa3baa3b2a2b2aa3b2a
11aba2abbab2a3a2babaab2b2aa3ba2baa2b2ab2aa3baa3b2a2b2aa3b2a
aaa2aba3a2babaab21a3ba2baa2b2ab2aba3baa3b2a2b2abab2a3b2ab2a
bbbab2ab2a3b2aabaab2aa2ba1a3ba2b2a2b2aa3b2aa3baaba3b2aa2a2b
a2a2a3a2b1a3ba2baa2b2aba3baa3b2a2b2aabbab2a3b2aabaab2b2aab2a
abababaab2a2b21ab2aa2baa2b2aa3baaba3b2a3b2ab2a2baa2bb2aa3a3b
babaab2a3ab2aa2ba1a3bba2b2aa3b2aa3bab2aba3b2aa2b2aabaa2ba2b2
b2b2b2aabaa3bab2aa2b21a3baa3b2ba2b2aaa2a3b2abaaba3a2bab2a2ba
a3a31a3baba3baa3b2a2abbab2a3b2aa2babaab2b2aa2baa2b2ab2aa2b2a
a2ba2ba2baa2b2a3b2aa2b2aa3baa3b2abaa2abb2b2aab2a3abaa3bab2a1b
abaabaa2b21a2b2aa3baababa3b2ab2a2baab2a2bb2aa3ab2aa2baa3ba3b2
ab2ab2ab2aa2baba2b2aa3b2abab2aba3b2aa2a3b2aabaa2b1a3ba2b2a3ba
b2ab2aa3bab2aa3baa3b2ba2b2ab2a2a3b2abaababaa3a2bab2a2b21a2baa
a3ba3ba3baa3b2b2a2a3b2abab2aabaa3a2bab2ab2aa2b21a2baba2b2aaab
a2baa2baa3b2aa3b2abaa2aba2bb2aab2a3abaa2b2a3bab2a1a2b2aa3babb2
a2b2a2b2a2b2aa3baaba3b2ab2a2abaab2a2bb2aa31ab2aa2baa3baba3b2ba
ab2aab2aba2b2abab2aba3b2aab2a3b2aabaa2ba2ba1a3ba2b2a3b2aa3baa2
a3baa3bab2a2b2aabaa3a2ba3bab2aa2b21a2baa3b2ba2b2aaa3b2abaabab2
a3b2a3b2a3b2abaa2bb2aab2a3a2baa2b2a3bab2a1aa2b2aa3baba2abb2aba
a2b2aa2b2aaba3b2aabaab2a2bb2aa2b21ab2aa2baa3ba3baaba3b2b2a2baa3
a3b2aa3b2aa2bb2aa2baa2b2a3bab2aa3b2aa2b2aa3babbaa2abb2ab2a3aba1

Right Cayley graph

Left Cayley graph

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. ba2 ⇒ ab2 [1]
2. bab ⇒ a3 [6]
3. b3 ⇒ aba [8]
4. a4 ⇒ 1 [5]
# ab:baa=abb,abab=1 ab - -
baa=abb
bab=aaa
bbb=aba
aaaa=1

Same cardinality

20 unique, 60 total

Σ#PresentationPropertiesDescriptionφ
8526⟨a, b | aab=a, bbbb=1⟩Noc 2019
92207⟨a, b | ab=aa, bbbb=b⟩Noc 201
104212⟨a, b | aaaaa=1, abbbb=1⟩Grp Com 20ℤ2016
105349⟨a, b | aaa=bb, bab=aa⟩Noc 201
106728⟨a, b | aba=a, aaaa=bb⟩Noc 20
106764⟨a, b | aba=b, aaaa=bb⟩Noc 20
107117⟨a, b | ab=aa, bbbb=bb⟩Noc 20
1112159⟨a, b | aaaa=aa, abbb=b⟩Noc 20
1112322⟨a, b | aaab=bb, bbba=a⟩Noc 20
1114367⟨a, b | aaaa=a, bbbbb=a⟩Noc Com 20ℕ(20 = 5)
1114407⟨a, b | aaaa=b, bbbbb=a⟩Noc Com 20ℕ(20 = 1)
1114408⟨a, b | aaaa=b, bbbbb=b⟩Noc Com 20ℕ(20 = 4)
1115933⟨a, b | aba=bb, baaab=a⟩Noc 20
1116124⟨a, b | aab=aa, baba=bb⟩Noc 20
1116339⟨a, b | aba=aa, aaaa=bb⟩Noc 20
1116343⟨a, b | aba=aa, aaab=bb⟩Noc 20
1116545⟨a, b | aba=bb, abb=aaa⟩Noc 20
1118619⟨a, b | aba=b, aaaaabb=1⟩Grp 203
1119503⟨a, b | aab=a, bbbbb=bb⟩Noc 20
1121047⟨a, b | ab=aa, bbbb=bbb⟩Noc 20

Isomorphic instances

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

5 total

Σ#PresentationMapping
105907⟨a, b | abab=1, aabba=b⟩φ(a) = aab, φ(b) = aaa
105915⟨a, b | abab=1, abbaa=b⟩φ(a) = a, φ(b) = b
1115164⟨a, b | aab=ba, abbabb=1⟩φ(a) = aaab, φ(b) = a
1115182⟨a, b | aab=ba, babbab=1⟩φ(a) = aaab, φ(b) = a
1115191⟨a, b | aab=ba, bbabba=1⟩φ(a) = aaab, φ(b) = a