#16339 ⟨a, b | aba=aa, aaaa=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.

1aba2abbaa3a2bba2baba4a3bba3ba2ba5ba4ba3ba6a7a8
11aba2abbaa3a2bba2baba4a3bba3ba2ba5ba4ba3ba6a7a8
aaa2aba3a2ba2a4a3ba3a2ba5ba4a4a3ba6a5ba4a7a8a5
bbbaa4ba2baba5ba3ba2ba6a5ba4ba3ba7a6a5a8a7a6a7a8
a2a2a3a2ba4a3ba3a5ba4a4a3ba6a5a5ba4a7a6a5a8a5a6
ababa2a5a3a2ba6a4a3ba7a6a5ba4a8a7a6a5a8a7a8a5
bababa2babba3ba2bba2ba4ba3bba3ba2ba5a8ba4ba3ba6a5a8a7a8a5
a3a3a4a3ba5ba4a4a6a5a5ba4a7a6a6a5a8a7a6a5a6a7
a2ba2ba3a6a4a3ba7a5ba4a8a7a6a5a5a8a7a6a5a8a5a6
ba2ba2ba3ba2bba4ba3bba3a5a8ba4ba3ba6a5a5a8a7a6a5a8a5a6
babbabba2a5ba3ba2ba6ba4ba3ba7a6a5a8a8a7a6a5a8a7a8a5
a4a4a5ba4a6a5a5a7a6a6a5a8a7a7a6a5a8a7a6a7a8
a3ba3ba4a7a5ba4a8a6a5a5a8a7a6a6a5a8a7a6a5a6a7
ba3ba3ba4ba3ba5a8ba4a6a5a5a8a7a6a6a5a8a7a6a5a6a7
ba2bba2bba3a6ba4ba3ba7a5a8a8a7a6a5a5a8a7a6a5a8a5a6
a5a5a6a5a7a6a6a8a7a7a6a5a8a8a7a6a5a8a7a8a5
ba4ba4a5a8a6a5a5a7a6a6a5a8a7a7a6a5a8a7a6a7a8
ba3bba3bba4a7a5a8a8a6a5a5a8a7a6a6a5a8a7a6a5a6a7
a6a6a7a6a8a7a7a5a8a8a7a6a5a5a8a7a6a5a8a5a6
a7a7a8a7a5a8a8a6a5a5a8a7a6a6a5a8a7a6a5a6a7
a8a8a5a8a6a5a5a7a6a6a5a8a7a7a6a5a8a7a6a7a8

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. a9 ⇒ a5 [4]
2. ba5 ⇒ a5 [5]
3. aba ⇒ a2 [1]
4. a4b ⇒ ba4 [3]
5. b2 ⇒ a4 [2]
# ab:aba=aa,aaaa=bb a/b - -
aaaaaaaaa=aaaaa
baaaaa=aaaaa
aba=aa
aaaab=baaaa
bb=aaaa

Same cardinality

20 unique, 65 total

Σ#PresentationPropertiesDescriptionφ
8526⟨a, b | aab=a, bbbb=1⟩Noc 2019
92207⟨a, b | ab=aa, bbbb=b⟩Noc 201
104095⟨a, b | baa=abb, abab=1⟩Grp 205
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
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