#5309 ⟨a, b | aaa=aa, 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.
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.

1aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab6a2
11aba2bab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab6a2
aaa2b3a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
bbbab2ba2b2ab3b2a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab5a2
a2a2a2b5a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
bababa2b4ba2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b2b2b2ab3b2a2b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
ba2ba2ba2b6ba2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b2ab2ab2a2b5b2a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b3b3b3ab4b3a2b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b2a2b2a2b2a2b5b2a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b3ab3ab3a2b6b3a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b4b4b4ab5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b3a2b3a2b3a2b6b3a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b4ab4ab4a2b5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b5b5b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b4a2b4a2b4a2b5b4a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b5ab5ab5a2b6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b6b6b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b5a2b5a2b5a2b6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab5a2
b6ab6ab6a2b5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2
b6a2b6a2b6a2b5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab5b6a2b5ab6b5a2b6ab6a2

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 ⇒ b5 [3]
2. ab ⇒ b3 [2]
3. a3 ⇒ a2 [1]
# ab:aaa=aa,bbb=ab b/a - -
bbbbbbb=bbbbb
ab=bbb
aaa=aa

Same cardinality

8 unique, 64 total

Σ#PresentationPropertiesDescriptionφ
8402⟨a, b | aaa=ab, bbb=1⟩Noc 2139
91599⟨a, b | aaa=ab, bbb=b⟩Noc 211
106265⟨a, b | aaa=a, abbbb=b⟩Noc 212
1111127⟨a, b | aaaaa=b, abbbb=1⟩Grp Com 21ℤ2113
1113091⟨a, b | bab=aaa, abba=b⟩Noc 21
1113248⟨a, b | bab=aaa, bbb=aa⟩Noc 21
1115158⟨a, b | aab=ba, ababab=1⟩Grp 211
1119212⟨a, b | aba=b, baaaab=a⟩Noc 21