#18831 ⟨a, b | aaa=b, bbbbbb=b⟩

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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.

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a3
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a3a4
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a3a4a5
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a17a3a4a5a6
a5a5a6a7a8a9a10a11a12a13a14a15a16a17a3a4a5a6a7
a6a6a7a8a9a10a11a12a13a14a15a16a17a3a4a5a6a7a8
a7a7a8a9a10a11a12a13a14a15a16a17a3a4a5a6a7a8a9
a8a8a9a10a11a12a13a14a15a16a17a3a4a5a6a7a8a9a10
a9a9a10a11a12a13a14a15a16a17a3a4a5a6a7a8a9a10a11
a10a10a11a12a13a14a15a16a17a3a4a5a6a7a8a9a10a11a12
a11a11a12a13a14a15a16a17a3a4a5a6a7a8a9a10a11a12a13
a12a12a13a14a15a16a17a3a4a5a6a7a8a9a10a11a12a13a14
a13a13a14a15a16a17a3a4a5a6a7a8a9a10a11a12a13a14a15
a14a14a15a16a17a3a4a5a6a7a8a9a10a11a12a13a14a15a16
a15a15a16a17a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
a16a16a17a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a3
a17a17a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a3a4

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. a18 ⇒ a3 [2]
2. b ⇒ a3 [1]
# ab:aaa=b,bbbbbb=b a/b - -
aaaaaaaaaaaaaaaaaa=aaa
b=aaa

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
1118758⟨a, b | aaa=a, bbbbbb=a⟩Noc Com 18ℕ(18 = 6)
1118830⟨a, b | aaa=b, bbbbbb=a⟩Noc Com 18ℕ(18 = 1)
1119624⟨a, b | aab=b, bbbba=aa⟩Noc 18
1120914⟨a, b | bb=aa, ababa=aa⟩Noc 181