#7117 ⟨a, b | ab=aa, bbbb=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.

1aba2bab2a3ba2b2ab3a4ba3b2a2b3aba4b2a3b3a2b2a4b3a3b3a4
11aba2bab2a3ba2b2ab3a4ba3b2a2b3aba4b2a3b3a2b2a4b3a3b3a4
aaa2a2a3a3a3a4a4a4a4a3a3a3a3a4a4a4a3a3a4
bbbab2ba2b2ab3ba3b2a2b3ab2ba4b2a3b3a2b2ab2a4b3a3b2a2b3a4b2a3b2a4
a2a2a3a3a4a4a4a3a3a3a3a4a4a4a4a3a3a3a4a4a3
bababa2ba2ba3ba3ba3ba4ba4ba4ba4ba3ba3ba3ba3ba4ba4ba4ba3ba3ba4
b2b2b2ab3b2a2b3ab2b2a3b3a2b2ab3b2a4b3a3b2a2b3ab3a4b2a3b3a2b2a4b3a3b3a4
a3a3a4a4a3a3a3a4a4a4a4a3a3a3a3a4a4a4a3a3a4
ba2ba2ba3ba3ba4ba4ba4ba3ba3ba3ba3ba4ba4ba4ba4ba3ba3ba3ba4ba4ba3
b2ab2ab2a2b2a2b2a3b2a3b2a3b2a4b2a4b2a4b2a4b2a3b2a3b2a3b2a3b2a4b2a4b2a4b2a3b2a3b2a4
b3b3b3ab2b3a2b2ab3b3a3b2a2b3ab2b3a4b2a3b3a2b2ab2a4b3a3b2a2b3a4b2a3b2a4
a4a4a3a3a4a4a4a3a3a3a3a4a4a4a4a3a3a3a4a4a3
ba3ba3ba4ba4ba3ba3ba3ba4ba4ba4ba4ba3ba3ba3ba3ba4ba4ba4ba3ba3ba4
b2a2b2a2b2a3b2a3b2a4b2a4b2a4b2a3b2a3b2a3b2a3b2a4b2a4b2a4b2a4b2a3b2a3b2a3b2a4b2a4b2a3
b3ab3ab3a2b3a2b3a3b3a3b3a3b3a4b3a4b3a4b3a4b3a3b3a3b3a3b3a3b3a4b3a4b3a4b3a3b3a3b3a4
ba4ba4ba3ba3ba4ba4ba4ba3ba3ba3ba3ba4ba4ba4ba4ba3ba3ba3ba4ba4ba3
b2a3b2a3b2a4b2a4b2a3b2a3b2a3b2a4b2a4b2a4b2a4b2a3b2a3b2a3b2a3b2a4b2a4b2a4b2a3b2a3b2a4
b3a2b3a2b3a3b3a3b3a4b3a4b3a4b3a3b3a3b3a3b3a3b3a4b3a4b3a4b3a4b3a3b3a3b3a3b3a4b3a4b3a3
b2a4b2a4b2a3b2a3b2a4b2a4b2a4b2a3b2a3b2a3b2a3b2a4b2a4b2a4b2a4b2a3b2a3b2a3b2a4b2a4b2a3
b3a3b3a3b3a4b3a4b3a3b3a3b3a3b3a4b3a4b3a4b3a4b3a3b3a3b3a3b3a3b3a4b3a4b3a4b3a3b3a3b3a4
b3a4b3a4b3a3b3a3b3a4b3a4b3a4b3a3b3a3b3a3b3a3b3a4b3a4b3a4b3a4b3a3b3a3b3a3b3a4b3a4b3a3

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 ⇒ a2 [1]
2. b4 ⇒ b2 [2]
3. a5 ⇒ a3 [3]
# ab:ab=aa,bbbb=bb ab - -
ab=aa
bbbb=bb
aaaaa=aaa

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
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