#7153 ⟨a, b | bb=aa, abab=ab⟩

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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
aaa2aba3a2babaa4a3ba2baaba5a4ba3abaa6a4a4b
bbbaa2a2bbaba3a2baa4(ba)2a3ba4ba2ba5a3a4a6a5
a2a2a3a2ba4a3ba2baa5a4ba3a2ba6a4a4a2baa4ba5a4
abababaa3a3baba4a3a5abaa4ba4a3ba6a4a5a4ba6
babaa2bbaba2baa4(ba)2a4ba2ba5baba4a6a2ba(ba)2a5a4ba6
a3a3a4a3ba5a4ba3a6a4a4a3ba4ba5a5a3a4a6a5
a2ba2ba2baa4a4ba2ba5a4a6a2baa4a5a4ba4ba5a6a4a4b
abaabaa3baba3a5abaa4a3ba6aba5a4ba3abaa6a4a4b
babbab(ba)2a2baa2bbaba4ba2baa4(ba)2a6a4ba2ba5a4ba4a6a5
a4a4a5a4ba6a4a4a4ba5a5a4ba4a6a6a4a5a4ba6
a3ba3ba3a5a4a3ba6a5a4ba3a5a6a4a4a6a4ba5a4
a2baa2baa4ba2ba4a6a2baa5a4ba4ba2ba6a4a4a2baa4ba5a4
(ba)2(ba)2a2bbaba2baa4(ba)2a4ba2ba5baba4a6a2ba(ba)2a5a4ba6
a5a5a6a4a4ba5a5a4a6a6a4a5a4ba4ba5a6a4a4b
a4ba4ba4a6a5a4ba4ba6a4a4a6a4ba5a5a4ba4a6a5
a6a6a4ba5a4a6a6a5a4ba4ba5a6a4a4a6a4ba5a4

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 ⇒ ab [2]
4. a3ba ⇒ a3 [4]
5. a5b ⇒ a4 [5]
6. a7 ⇒ a4b [6]
# ab:bb=aa,abab=ab ab - -
bb=aa
baa=aab
abab=ab
aaaba=aaa
aaaaab=aaaa
aaaaaaa=aaaab

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
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