#1022 ⟨a, b | ab=a, bbb=ba⟩

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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
  8. Isomorphic instances

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.

1abbab2
11abbab2
aaaaaa
bbbab2baba
babababababa
b2b2babababa

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. a2 ⇒ a [3]
2. ab ⇒ a [1]
3. b2a ⇒ ba [4]
4. b3 ⇒ ba [2]
# ab:ab=a,bbb=ba ab - -
aa=a
ab=a
bba=ba
bbb=ba

Same cardinality

11 unique, 1458 total

Σ#PresentationPropertiesDescriptionφ
644⟨a, b | aa=b, abb=1⟩Grp Com 5ℤ51132
7253⟨a, b | aa=b, abb=a⟩Noc Com 5ℕ(5 = 1)71
7254⟨a, b | aa=b, abb=b⟩Noc Com 5ℕ(5 = 2)43
7268⟨a, b | ab=a, baa=b⟩Noc 586
8574⟨a, b | aaa=b, aab=b⟩Noc Com 5ℕ(5 = 3)27
8950⟨a, b | ab=a, bbbb=a⟩Noc Com 5ℕ(5 = 4)32
8995⟨a, b | ab=a, aaa=bb⟩Noc Com 519
81019⟨a, b | ab=a, bba=bb⟩Noc 525
81020⟨a, b | ab=a, bbb=aa⟩Noc Com 59
108617⟨a, b | aa=a, abbbba=b⟩Noc Com 53
1115426⟨a, b | aaa=aa, abbba=b⟩Noc Com 5

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

4 total

Σ#PresentationMapping
93198⟨a, b | ab=a, bbb=bab⟩φ(a) = a, φ(b) = b
109351⟨a, b | ab=a, babb=bbb⟩φ(a) = a, φ(b) = b
1120015⟨a, b | aab=a, babb=bbb⟩φ(a) = a, φ(b) = b
1125595⟨a, b | ab=a, babbb=bbb⟩φ(a) = a, φ(b) = b