#18758 ⟨a, b | aaa=a, bbbbbb=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.

1bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17
11bb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17
bbb2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b6
b2b2b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b6b7
b3b3b4b5b6b7b8b9b10b11b12b13b14b15b16b17b6b7b8
b4b4b5b6b7b8b9b10b11b12b13b14b15b16b17b6b7b8b9
b5b5b6b7b8b9b10b11b12b13b14b15b16b17b6b7b8b9b10
b6b6b7b8b9b10b11b12b13b14b15b16b17b6b7b8b9b10b11
b7b7b8b9b10b11b12b13b14b15b16b17b6b7b8b9b10b11b12
b8b8b9b10b11b12b13b14b15b16b17b6b7b8b9b10b11b12b13
b9b9b10b11b12b13b14b15b16b17b6b7b8b9b10b11b12b13b14
b10b10b11b12b13b14b15b16b17b6b7b8b9b10b11b12b13b14b15
b11b11b12b13b14b15b16b17b6b7b8b9b10b11b12b13b14b15b16
b12b12b13b14b15b16b17b6b7b8b9b10b11b12b13b14b15b16b17
b13b13b14b15b16b17b6b7b8b9b10b11b12b13b14b15b16b17b6
b14b14b15b16b17b6b7b8b9b10b11b12b13b14b15b16b17b6b7
b15b15b16b17b6b7b8b9b10b11b12b13b14b15b16b17b6b7b8
b16b16b17b6b7b8b9b10b11b12b13b14b15b16b17b6b7b8b9
b17b17b6b7b8b9b10b11b12b13b14b15b16b17b6b7b8b9b10

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. b18 ⇒ b6 [4]
2. a ⇒ b6 [2]
# ab:aaa=a,bbbbbb=a b/a - -
bbbbbbbbbbbbbbbbbb=bbbbbb
a=bbbbbb

Same cardinality

17 unique, 104 total

Σ#PresentationPropertiesDescriptionφ
81121⟨a, b | aa=1, abbba=b⟩Noc 1817
93387⟨a, b | aa=1, abbbbb=b⟩Noc 1831
105521⟨a, b | aaab=1, bbbbbb=1⟩Grp Com 18ℤ1833
106732⟨a, b | aba=a, aaab=bb⟩Noc 18
106788⟨a, b | aba=b, baab=aa⟩Noc 18
106795⟨a, b | aba=b, bbbb=aa⟩Noc 18
107039⟨a, b | bb=aa, aaaba=b⟩Noc 182
108910⟨a, b | aa=a, bbbbb=ab⟩Noc 181
1112187⟨a, b | aaaa=ab, babb=b⟩Noc 182
1115797⟨a, b | aab=bb, bbbba=a⟩Noc 18
1116042⟨a, b | aaa=ab, bbbb=ab⟩Noc 18
1116043⟨a, b | aaa=ab, bbbb=ba⟩Noc 18
1116313⟨a, b | aab=bb, baba=aa⟩Noc 18
1118830⟨a, b | aaa=b, bbbbbb=a⟩Noc Com 18ℕ(18 = 1)
1118831⟨a, b | aaa=b, bbbbbb=b⟩Noc Com 18ℕ(18 = 3)
1119624⟨a, b | aab=b, bbbba=aa⟩Noc 18
1120914⟨a, b | bb=aa, ababa=aa⟩Noc 181