#15933 ⟨a, b | aba=bb, baaab=a⟩

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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.

1aba2abbaa3ababa2a4aba2ba3a5aba3ba4a6aba4ba5aba5ba6
11aba2abbaa3ababa2a4aba2ba3a5aba3ba4a6aba4ba5aba5ba6
aaa2aba3ba2abaa4ba3aba2a5ba4aba3a6ba5aba4aba6aba5baab
bbbaababa2a5aba2ba3a6aba3ba4aaba4ba5a2aba5ba6a3aba4aba
a2a2a3ba2a4aba2ba3a5aba3ba4a6aba4ba5aaba5ba6a2abbaababa2
ababababa3aba2a6ba4aba3aba5aba4a2ba6aba5a3baaba4ba2a5ba3
bababa2a5ba3aba3a6ba4aba4aba5aba5a2ba6aba3baabaa4aba2a5
a3a3a4aba2a5ba4aba3a6ba5aba4aba6aba5a2baaba3ba2ababa3aba2
abaabaaba2a6aba3ba5aaba4ba6a2aba5baa3abba2a4ababa3a5ba4a6
ba2ba2ba3aba3ba4aaba4ba5a2aba5ba6a3abbaa4ababa2a5aba2a6aba3
a4a4a5ba4a6aba4ba5aaba5ba6a2abbaa3ababa2a4aba2ba3aba3ba4
aba2aba2aba3ba5aba4a2ba6aba5a3baaba4ba2abaa5ba3aba2a6ba4aba5
ba3ba3ba4aba5aba5a2ba6aba3baabaa4ba2aba2a5ba3aba3a6aba4a
a5a5a6aba4aba6aba5a2baaba3ba2abaa4ba3aba2a5ba4aba3ba5aba4
aba3aba3aba4a2aba5baa3abba2a4ababa3a5aba2ba4a6aba3ba5aba6a2
ba4ba4ba5aba5ba6a3abbaa4ababa2a5aba2ba3a6aba3ba4aaba4a2aba5
a6a6aba6a2abbaa3ababa2a4aba2ba3a5aba3ba4a6aba4ba5aba5ba6
aba4aba4aba5baaba4ba2abaa5ba3aba2a6ba4aba3aba5aba4a2ba6a3ba
ba5ba5ba6a3baabaa4ba2aba2a5ba3aba3a6ba4aba4aba5aba5a2aba3
aba5aba5aba4ababa3a5aba2ba4a6aba3ba5aaba4ba6a2aba5baa3ba2a4
ba6ba6baababa2a5aba2ba3a6aba3ba4aaba4ba5a2aba5ba6a3aba4aba

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. a7 ⇒ a [7]
2. aba6 ⇒ ab [6]
3. a2b ⇒ ba2 [10]
4. b2 ⇒ aba [1]
5. bab ⇒ a5 [11]
# ab:aba=bb,baaab=a a/b - -
aaaaaaa=a
abaaaaaa=ab
aab=baa
bb=aba
bab=aaaaa

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)
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