#12322 ⟨a, b | aaab=bb, bbba=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

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.

1aba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8ba9b
11aba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8ba9b
aaa2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9bab
bba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba3b
a2a2a3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba2b
ababa5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba4b
a3a3a4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba3b
a2ba2ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba5b
a4a4a5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba4b
a3ba3ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba6b
a5a5a6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba5b
a4ba4ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
a6a6a7a6ba8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba6b
a5ba5ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba8b
a7a7a8a7ba9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba7b
a6ba6baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8ba9b
a8a8a9a8baa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba8b
a7ba7ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9bab
a9a9aa9ba2aba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8ba9b
a8ba8ba3a2ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba2b
a9ba9ba4a3ba5a4ba6a5ba7a6ba8a7ba9a8baa9ba2aba3a2ba3b

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. a10 ⇒ a [8]
2. ba ⇒ a4 [7]
3. b2 ⇒ a3b [1]
# ab:aaab=bb,bbba=a a/b - -
aaaaaaaaaa=a
ba=aaaa
bb=aaab

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
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