#7152 ⟨a, b | bb=aa, abab=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.

1aba2abbaa3a2babababa4a3ba2ba(ba)2a5a4ba6
11aba2abbaa3a2babababa4a3ba2ba(ba)2a5a4ba6
aaa2aba3a2babaa4a3ba2baa2a5a4ba2ba3a6a2baa3
bbbaa2a2bbaba3a2baa4(ba)2a3ba4ba3a5a2ba3ba6a2b
a2a2a3a2ba4a3ba2baa5a4ba2ba3a6a2baa3ba4a3a2ba4
abababaa3a3ba2a4a2ba5a3a4ba2baa4a6a3ba4ba3a3b
babaa2bbaba2baa4(ba)2a4ba3a5a2ba3ba6a4a2baa2ba5a2ba
a3a3a4a3ba5a4ba2ba6a2baa3ba4a3a2ba4ba5a4a3ba5
a2ba2ba2baa4a4ba3a5a3ba6a4a2baa2ba5a3a4ba2baa4a4b
abaabaa3ba2a2ba5a3a2baa4a6a3ba4ba3a5a2ba3ba6a2b
babbab(ba)2a2baa3a2ba4ba4a3ba2baa6a5a4ba2ba3a6a2baa3
a4a4a5a4ba6a2baa3ba3a2ba4ba5a4a3ba2baa6a5a4ba6
a3ba3ba2ba5a2baa4a6a4ba3a5a2ba3ba6a4a2baa2ba5a2ba
a2baa2baa4ba3a3ba6a4a2ba5a3a4ba2baa4a6a3ba4ba3a3b
(ba)2(ba)2a3a2ba4a3ba2baa5a4ba2ba3a6a2baa3ba4a3a2ba4
a5a5a6a2baa3a2ba4ba4a3ba2baa6a5a4ba2ba3a6a2baa3
a4ba4ba3ba6a2ba5a3a2baa4a6a3ba4ba3a5a2ba3ba6a2b
a6a6a3a2ba4a3ba2baa5a4ba2ba3a6a2baa3ba4a3a2ba4

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. b2 ⇒ a2 [1]
2. ba2 ⇒ a2b [3]
3. (ab)2 ⇒ a2 [2]
4. a3ba ⇒ a2b [4]
5. a5b ⇒ a2ba [5]
6. a7 ⇒ a3 [6]
# ab:bb=aa,abab=aa ab - -
bb=aa
baa=aab
abab=aa
aaaba=aab
aaaaab=aaba
aaaaaaa=aaa

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
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