#15426 ⟨a, b | aaa=aa, abbba=b⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality

Properties

Commutative structure

Finite structure

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.

Staircase diagram

Cayley table

Idempotents are shown in bold.

1aba2b2
11aba2b2
aaa2ba2b2
bbbb2bb
a2a2a2ba2b2
b2b2b2bb2b2

Right 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. ab ⇒ b [5]
2. ba ⇒ b [7]
3. a3 ⇒ a2 [1]
4. b3 ⇒ b [8]
# ab:aaa=aa,abbba=b ab - -
ab=b
ba=b
aaa=aa
bbb=b

Same cardinality

11 unique, 1462 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
81022⟨a, b | ab=a, bbb=ba⟩Noc 54
108617⟨a, b | aa=a, abbbba=b⟩Noc Com 53