#7154 ⟨a, b | bb=aa, abab=ba⟩

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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.

1aba2abbaa3ababa2a4aba2ba3a5aba3ba4a6aba4
11aba2abbaa3ababa2a4aba2ba3a5aba3ba4a6aba4
aaa2aba3ba2abaa4ba3aba2a5ba4aba3a6baaba4a3ba2
bbbaa2ba2aba3a3ba3aba4a4ba4abaa5baaba2a6ba2aba3
a2a2a3ba2a4aba2ba3a5aba3ba4a6aba4baa3ababa2a4aba2
abababaa3aba2baa4aba3ba2a5aba4ba3a6ababa4a3aba2ba
bababa2aba3ba3a4aba4ba4a5ababaa6aba2ba2a3aba3ba3a4
a3a3a4aba2a5ba4aba3a6baaba4a3ba2abaa4ba3aba2a5ba4
abaabaaba2baaba3a5ba2aba4a6ba3abaa3ba4aba2a4baaba3a5
ba2ba2ba3a4ba4abaa5baaba2a6ba2aba3a3ba3aba4a4ba4aba
a4a4a5ba4a6aba4baa3ababa2a4aba2ba3a5aba3ba4a6aba4
aba2aba2aba3a5aba4ba3a6ababa4a3aba2baa4aba3ba2a5aba4ba3
ba3ba3ba4ababaa6aba2ba2a3aba3ba3a4aba4ba4a5ababaa6
a5a5a6aba4a3ba2abaa4ba3aba2a5ba4aba3a6baaba4a3ba2
aba3aba3aba4ba3abaa3ba4aba2a4baaba3a5ba2aba4a6ba3abaa3
ba4ba4baa6ba2aba3a3ba3aba4a4ba4abaa5baaba2a6ba2aba3
a6a6a3ba2a4aba2ba3a5aba3ba4a6aba4baa3ababa2a4aba2
aba4aba4abaa3aba2baa4aba3ba2a5aba4ba3a6ababa4a3aba2ba

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 ⇒ a3 [6]
2. ba5 ⇒ ba [7]
3. a2b ⇒ ba2 [3]
4. b2 ⇒ a2 [1]
5. bab ⇒ aba3 [4]
# ab:bb=aa,abab=ba a/b - -
aaaaaaa=aaa
baaaaa=ba
aab=baa
bb=aa
bab=abaaa

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
106514⟨a, b | aba=b, baaab=a⟩Noc 17
107152⟨a, b | bb=aa, abab=aa⟩Noc 17
107153⟨a, b | bb=aa, abab=ab⟩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