#6788 ⟨a, b | aba=b, baab=aa⟩

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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.

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9
aaa2ba7a3ba4baa5ba2a6ba3a7ba4a8ba5a9ba6a2
bbbaa4ba2a5ba3a6ba4a7ba5a8ba6a9ba7a2ba3ba
a2a2a3ba6a4ba7a5ba6baa7ba2a8ba3a9ba4a2ba5a3
bababa2a3ba3a4ba4a5ba5a6ba6a7ba7a8ba9baa2ba2
a3a3a4ba5a5ba6a6ba7a7ba8baa9ba2a2ba3a3ba4a4
ba2ba2ba3a2ba4a3ba5a4ba6a5ba7a6ba7baa8ba2a9ba3
a4a4a5ba4a6ba5a7ba6a8ba7a9ba2baa3ba2a4ba3a5
ba3ba3ba4a9ba5a2ba6a3ba7a4ba5baa6ba2a7ba3a8ba4
a5a5a6ba3a7ba4a8ba5a9ba6a2ba7a3ba4baa5ba2a6
ba4ba4ba5a8ba6a9ba7a2ba3baa4ba2a5ba3a6ba4a7ba5
a6a6a7ba2a8ba3a9ba4a2ba5a3ba6a4ba7a5ba6baa7
ba5ba5ba6a7ba7a8ba9baa2ba2a3ba3a4ba4a5ba5a6ba6
a7a7a8baa9ba2a2ba3a3ba4a4ba5a5ba6a6ba7a7ba8
ba6ba6ba7a6ba7baa8ba2a9ba3a2ba4a3ba5a4ba6a5ba7
a8a8a9ba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7a9
ba7ba7ba5baa6ba2a7ba3a8ba4a9ba5a2ba6a3ba7a4b
a9a9a2ba7a3ba4baa5ba2a6ba3a7ba4a8ba5a9ba6a2

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. a10 ⇒ a2 [8]
2. ba8 ⇒ b [7]
3. ab ⇒ ba7 [6]
4. b2 ⇒ a4 [4]
# ab:aba=b,baab=aa a/b - -
aaaaaaaaaa=aa
baaaaaaaa=b
ab=baaaaaaa
bb=aaaa

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
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)
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