#16545 ⟨a, b | aba=bb, abb=aaa⟩

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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.

1aba2abbaa3a2bababa2baba4a3baba2(ab)2ba3ba2ba5aba3aba2b
11aba2abbaa3a2bababa2baba4a3baba2(ab)2ba3ba2ba5aba3aba2b
aaa2aba3a2babaa4a3ba3aba2(ab)2a5a4a4a3baba3aba2ba4a5a4
bbbaababa2bababa2ba3ba2b(ab)2aba3aba2ba4a5a3baba3a5a4a5a4a5
a2a2a3a2ba4a3ba3a5a4a4a4a3ba4a5a5a4a5a4a5a4a5
abababaa3aba2(ab)2a4aba3aba2ba3ba5a4a5a4a4a5a4a5a4a5a4
bababa2babba3ba2b(ab)2a4a5ba3a3baba3a5a4a4a5a4a5a4a5a4
a3a3a4a3ba5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
a2ba2ba3a4a4a3ba5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
abaabaaba2(ab)2aba3aba2ba3ba5a4aba3a4a5a4a5a5a4a5a4a5a4a5
ba2ba2ba3ba2ba4a5ba3a5a4a4a4a5a4a5a5a4a5a4a5a4a5
babbab(ab)2ba3a3baba3a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
a4a4a5a4a4a5a5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
a3ba3ba4a5a5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
aba2aba2aba3aba2ba5a4aba3a4a5a5a5a4a5a4a4a5a4a5a4a5a4
(ab)2(ab)2a3baba3a4a5a5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
ba3ba3a4a5a5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
ba2bba2bba3a4a4a5a5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
a5a5a4a5a5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4
aba3aba3a5a4a4a5a5a5a4a4a4a5a4a5a5a4a5a4a5a4a5
aba2baba2baba3a5a5a4a4a4a5a5a5a4a5a4a4a5a4a5a4a5a4

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. a6 ⇒ a4 [5]
2. a4b ⇒ a4 [4]
3. a2ba ⇒ a3 [2]
4. ba4 ⇒ a4 [7]
5. b2 ⇒ aba [1]
6. ba3b ⇒ a5 [8]
7. (ba)2 ⇒ (ab)2 [3]
# ab:aba=bb,abb=aaa reversed:a/b - -
aaaaaa=aaaa
aaaab=aaaa
aaba=aaa
baaaa=aaaa
bb=aba
baaab=aaaaa
baba=abab

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)
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
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