#6764 ⟨a, b | aba=b, 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.

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9a10a11
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9a10a11
aaa2ba7a3ba4baa5ba2a6ba3a7ba4a8ba5a9ba6a10a11a4
bbbaa4ba2a5ba3a6ba4a7ba5a8ba6a9ba7a10ba11baba2ba3
a2a2a3ba6a4ba7a5ba6baa7ba2a8ba3a9ba4a10ba5a11a4a5
bababa2a11ba3a4ba4a5ba5a6ba6a7ba7a8ba9baa10ba2ba3ba4
a3a3a4ba5a5ba6a6ba7a7ba8baa9ba2a10ba3a11ba4a4a5a6
ba2ba2ba3a10ba4a11ba5a4ba6a5ba7a6ba7baa8ba2a9ba3ba4ba5
a4a4a5ba4a6ba5a7ba6a8ba7a9ba10baa11ba2a4ba3a5a6a7
ba3ba3ba4a9ba5a10ba6a11ba7a4ba5baa6ba2a7ba3a8ba4ba5ba6
a5a5a6ba3a7ba4a8ba5a9ba6a10ba7a11ba4baa5ba2a6a7a8
ba4ba4ba5a8ba6a9ba7a10ba11baa4ba2a5ba3a6ba4a7ba5ba6ba7
a6a6a7ba2a8ba3a9ba4a10ba5a11ba6a4ba7a5ba6baa7a8a9
ba5ba5ba6a7ba7a8ba9baa10ba2a11ba3a4ba4a5ba5a6ba6ba7b
a7a7a8baa9ba2a10ba3a11ba4a4ba5a5ba6a6ba7a7ba8a9a10
ba6ba6ba7a6ba7baa8ba2a9ba3a10ba4a11ba5a4ba6a5ba7bba
a8a8a9ba10baa11ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9a10a11
ba7ba7ba5baa6ba2a7ba3a8ba4a9ba5a10ba6a11ba7a4bbaba2
a9a9a10ba7a11ba4baa5ba2a6ba3a7ba4a8ba5a9ba6a10a11a4
a10a10a11ba6a4ba7a5ba6baa7ba2a8ba3a9ba4a10ba5a11a4a5
a11a11a4ba5a5ba6a6ba7a7ba8baa9ba2a10ba3a11ba4a4a5a6

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. a12 ⇒ a4 [8]
2. ba8 ⇒ b [7]
3. ab ⇒ ba7 [6]
4. b2 ⇒ a4 [2]
# ab:aba=b,aaaa=bb a/b - -
aaaaaaaaaaaa=aaaa
baaaaaaaa=b
ab=baaaaaaa
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
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