#13156 ⟨a, b | aba=aaa, bbb=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.
A right zero element x satisfies yx = 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.

1aba2bab2ba2b2ab3b2a2b3ab4b4ab5b5ab6b6a
11aba2bab2ba2b2ab3b2a2b3ab4b4ab5b5ab6b6a
aaa2b3b3ab3ab4b5ab4ab5b6ab5ab6b6ab6b6ab6b6a
bbbab2ba2b2ab3b2a2b3ab4b5ab4ab5b5ab6b6ab6b6a
a2a2b3ab5b5ab5ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
bababa2b4b4ab4ab5b6ab5ab6b6ab6ab6b6ab6b6ab6b6a
b2b2b2ab3b2a2b3ab4b5ab4ab5b6ab5ab6b6ab6b6ab6b6a
ba2ba2b4ab6b6ab6ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b2ab2ab2a2b5b5ab5ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b3b3b3ab4b5ab4ab5b6ab5ab6b6ab6ab6b6ab6b6ab6b6a
b2a2b2a2b5ab6b6ab6ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b3ab3ab5ab6b6ab6ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b4b4b4ab5b6ab5ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b4ab4ab6ab6b6ab6ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b5b5b5ab6b6ab6ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b5ab5ab6ab6b6ab6ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b6b6b6ab6b6ab6ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a
b6ab6ab6ab6b6ab6ab6b6ab6ab6b6ab6ab6b6ab6b6ab6b6a

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. b7 ⇒ b6 [5]
2. ab ⇒ b3 [2]
3. b3a2 ⇒ b5a [4]
4. a3 ⇒ b3a [3]
# ab:aba=aaa,bbb=ab b/a - -
bbbbbbb=bbbbbb
ab=bbb
bbbaa=bbbbba
aaa=bbba

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
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