#14367 ⟨a, b | aaaa=a, bbbbb=a⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality

Properties

Commutative structure

Finite structure

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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19
11bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19
bbb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5
b2b2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6
b3b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6b7
b4b4b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6b7b8
b5b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6b7b8b9
b6b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6b7b8b9b10
b7b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6b7b8b9b10b11
b8b8b9b10b11b12b13b14b15b16b17b18b19b5b6b7b8b9b10b11b12
b9b9b10b11b12b13b14b15b16b17b18b19b5b6b7b8b9b10b11b12b13
b10b10b11b12b13b14b15b16b17b18b19b5b6b7b8b9b10b11b12b13b14
b11b11b12b13b14b15b16b17b18b19b5b6b7b8b9b10b11b12b13b14b15
b12b12b13b14b15b16b17b18b19b5b6b7b8b9b10b11b12b13b14b15b16
b13b13b14b15b16b17b18b19b5b6b7b8b9b10b11b12b13b14b15b16b17
b14b14b15b16b17b18b19b5b6b7b8b9b10b11b12b13b14b15b16b17b18
b15b15b16b17b18b19b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19
b16b16b17b18b19b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5
b17b17b18b19b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6
b18b18b19b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6b7
b19b19b5b6b7b8b9b10b11b12b13b14b15b16b17b18b19b5b6b7b8

Right 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. b20 ⇒ b5 [4]
2. a ⇒ b5 [2]
# ab:aaaa=a,bbbbb=a b/a - -
bbbbbbbbbbbbbbbbbbbb=bbbbb
a=bbbbb

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
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