#14408 ⟨a, b | aaaa=b, bbbbb=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.

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4a5
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4a5a6
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4a5a6a7
a5a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4a5a6a7a8
a6a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4a5a6a7a8a9
a7a7a8a9a10a11a12a13a14a15a16a17a18a19a4a5a6a7a8a9a10
a8a8a9a10a11a12a13a14a15a16a17a18a19a4a5a6a7a8a9a10a11
a9a9a10a11a12a13a14a15a16a17a18a19a4a5a6a7a8a9a10a11a12
a10a10a11a12a13a14a15a16a17a18a19a4a5a6a7a8a9a10a11a12a13
a11a11a12a13a14a15a16a17a18a19a4a5a6a7a8a9a10a11a12a13a14
a12a12a13a14a15a16a17a18a19a4a5a6a7a8a9a10a11a12a13a14a15
a13a13a14a15a16a17a18a19a4a5a6a7a8a9a10a11a12a13a14a15a16
a14a14a15a16a17a18a19a4a5a6a7a8a9a10a11a12a13a14a15a16a17
a15a15a16a17a18a19a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18
a16a16a17a18a19a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19
a17a17a18a19a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4
a18a18a19a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4a5
a19a19a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a4a5a6

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. a20 ⇒ a4 [2]
2. b ⇒ a4 [1]
# ab:aaaa=b,bbbbb=b a/b - -
aaaaaaaaaaaaaaaaaaaa=aaaa
b=aaaa

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
1114367⟨a, b | aaaa=a, bbbbb=a⟩Noc Com 20ℕ(20 = 5)
1114407⟨a, b | aaaa=b, bbbbb=a⟩Noc Com 20ℕ(20 = 1)
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