#13141 ⟨a, b | aab=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.
The zero element z satisfies zy = yz = z 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.

1aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b6
11aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b6
aaa2b3b5b3ab4b3a2b4ab5b4a2b6b6b6b6b6b6b6
bbbab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b6b6b6b6
a2a2b5b5b6b6b6b6b6b6b6b6b6b6b6b6b6b6
bababa2b4b6b4ab5b4a2b6b6b6b6b6b6b6b6b6b6
b2b2b2ab3b2a2b3ab4b3a2b4ab5b4a2b6b6b6b6b6b6b6
ba2ba2b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
b2ab2ab2a2b5b6b6b6b6b6b6b6b6b6b6b6b6b6b6
b3b3b3ab4b3a2b4ab5b4a2b6b6b6b6b6b6b6b6b6b6
b2a2b2a2b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
b3ab3ab3a2b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
b4b4b4ab5b4a2b6b6b6b6b6b6b6b6b6b6b6b6b6
b3a2b3a2b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
b4ab4ab4a2b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
b5b5b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
b4a2b4a2b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6
b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6b6

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. b5a ⇒ b6 [6]
4. a3 ⇒ b5 [3]
# ab:aab=aaa,bbb=ab b/a - -
bbbbbbb=bbbbbb
ab=bbb
bbbbba=bbbbbb
aaa=bbbbb

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
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