#6514 ⟨a, b | aba=b, 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.

1aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7
11aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7
aaa2ba7a3ba4baa5ba2a6ba3a7ba4a8ba5aba6
bbbaa4ba2a5ba3a6ba4a7ba5a8ba6aba7a2ba3
a2a2a3ba6a4ba7a5ba6baa7ba2a8ba3aba4a2ba5
bababa2a3ba3a4ba4a5ba5a6ba6a7ba7a8babaa2
a3a3a4ba5a5ba6a6ba7a7ba8baaba2a2ba3a3ba4
ba2ba2ba3a2ba4a3ba5a4ba6a5ba7a6ba7baa8ba2a
a4a4a5ba4a6ba5a7ba6a8ba7aba2baa3ba2a4ba3
ba3ba3ba4aba5a2ba6a3ba7a4ba5baa6ba2a7ba3a8
a5a5a6ba3a7ba4a8ba5aba6a2ba7a3ba4baa5ba2
ba4ba4ba5a8ba6aba7a2ba3baa4ba2a5ba3a6ba4a7
a6a6a7ba2a8ba3aba4a2ba5a3ba6a4ba7a5ba6ba
ba5ba5ba6a7ba7a8babaa2ba2a3ba3a4ba4a5ba5a6
a7a7a8baaba2a2ba3a3ba4a4ba5a5ba6a6ba7a7b
ba6ba6ba7a6ba7baa8ba2aba3a2ba4a3ba5a4ba6a5
a8a8aba2baa3ba2a4ba3a5ba4a6ba5a7ba6a8ba7
ba7ba7ba5baa6ba2a7ba3a8ba4aba5a2ba6a3ba7a4

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. a9 ⇒ a [8]
2. ba8 ⇒ b [7]
3. ab ⇒ ba7 [6]
4. b2 ⇒ a4 [5]
# ab:aba=b,baaab=a a/b - -
aaaaaaaaa=a
baaaaaaaa=b
ab=baaaaaaa
bb=aaaa

Same cardinality

16 unique, 71 total

Σ#PresentationPropertiesDescriptionφ
104375⟨a, b | aaaa=b, abbbb=1⟩Grp Com 17ℤ1739
106428⟨a, b | aab=b, babbb=a⟩Noc 172
107152⟨a, b | bb=aa, abab=aa⟩Noc 17
107153⟨a, b | bb=aa, abab=ab⟩Noc 17
107154⟨a, b | bb=aa, abab=ba⟩Noc 17
1113141⟨a, b | aab=aaa, bbb=ab⟩Noc 17
1113142⟨a, b | aab=aaa, bbb=ba⟩Noc 17
1113151⟨a, b | aba=aaa, baa=bb⟩Noc 17
1113156⟨a, b | aba=aaa, bbb=ab⟩Noc 17
1113218⟨a, b | abb=aba, bbb=aa⟩Noc 17
1114395⟨a, b | aaaa=b, abbbb=a⟩Noc Com 17ℕ(17 = 1)2
1114396⟨a, b | aaaa=b, abbbb=b⟩Noc Com 17ℕ(17 = 4)2
1114698⟨a, b | aabb=a, baaaa=b⟩Noc 177
1115488⟨a, b | aaa=ab, babbb=b⟩Noc 173
1116143⟨a, b | aab=aa, bbbb=ba⟩Noc 17
1116521⟨a, b | aab=bb, bab=aaa⟩Noc 17