#16521 ⟨a, b | aab=bb, bab=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.

1aba2abbaa3a2bababa2a4a2baaba2a5a2ba2a6a7
11aba2abbaa3a2bababa2a4a2baaba2a5a2ba2a6a7
aaa2aba3a2babaa4a7a2baaba2a5a6a2ba2a6a7a7a6
bbbaa2bba2a3a2baa5a6a4a2ba2a6a7a5a7a6a6a7
a2a2a3a2ba4a7a2baa5a6a6a2ba2a6a7a7a7a6a6a7
abababaa7aba2a4a6a6a7a5a7a7a6a6a6a7a7a6
bababa2a3a5a6a4a6a7a7a5a7a6a6a6a7a7a6
a3a3a4a7a5a6a6a6a7a7a7a7a6a6a6a7a7a6
a2ba2ba2baa6a2ba2a5a7a7a6a6a6a6a7a7a7a6a6a7
abaabaaba2a4a6a7a5a7a6a6a6a6a7a7a7a6a6a7
ba2ba2a5a6a6a7a7a7a6a6a6a6a7a7a7a6a6a7
a4a4a5a6a6a7a7a7a6a6a6a6a7a7a7a6a6a7
a2baa2baa2ba2a5a7a6a6a6a7a7a7a7a6a6a6a7a7a6
aba2aba2a6a7a7a6a6a6a7a7a7a7a6a6a6a7a7a6
a5a5a6a7a7a6a6a6a7a7a7a7a6a6a6a7a7a6
a2ba2a2ba2a7a6a6a7a7a7a6a6a6a6a7a7a7a6a6a7
a6a6a7a6a6a7a7a7a6a6a6a6a7a7a7a6a6a7
a7a7a6a7a7a6a6a6a7a7a7a7a6a6a6a7a7a6

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. a8 ⇒ a6 [7]
2. ba3 ⇒ a5 [4]
3. a3b ⇒ a7 [9]
4. b2 ⇒ a2b [1]
5. bab ⇒ a3 [2]
6. ba2b ⇒ a6 [8]
# ab:aab=bb,bab=aaa a/b - -
aaaaaaaa=aaaaaa
baaa=aaaaa
aaab=aaaaaaa
bb=aab
bab=aaa
baab=aaaaaa

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
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