#13151 ⟨a, b | aba=aaa, baa=bb⟩

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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.
A left zero element x satisfies xy = x for all y.
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.

1aba2abbaa3a2bba2baba4a3bba3a5ba4ba3ba6
11aba2abbaa3a2bba2baba4a3bba3a5ba4ba3ba6
aaa2aba3a2ba3a4a3ba4a3ba5a6a5a6a6a6a6
bbbaba2ba2babba3ba3ba4ba4ba3bba4ba3bba4ba4ba4ba4ba4
a2a2a3a2ba4a3ba4a5a6a5a6a6a6a6a6a6a6a6
ababa3a4a4a3ba5a5a6a6a6a6a6a6a6a6a6a6
bababa2babba3ba4ba3ba4ba3bba4ba3bba4ba4ba4ba4ba4ba4ba4
a3a3a4a3ba5a6a5a6a6a6a6a6a6a6a6a6a6a6
a2ba2ba4a5a5a6a6a6a6a6a6a6a6a6a6a6a6a6
ba2ba2ba3ba4ba4ba3bba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
babbabba3ba4ba4ba3bba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
a4a4a5a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6
a3ba3ba5a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6
ba3ba3ba4ba3bba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
a5a5a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6
ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
ba3bba3bba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6a6

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 ⇒ a6 [6]
2. ba5 ⇒ ba4 [5]
3. aba ⇒ a3 [1]
4. a4b ⇒ a6 [4]
5. b2 ⇒ ba2 [2]
6. ba2b ⇒ ba4 [3]
# ab:aba=aaa,baa=bb a/b - -
aaaaaaa=aaaaaa
baaaaa=baaaa
aba=aaa
aaaab=aaaaaa
bb=baa
baab=baaaa

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
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