#11133 ⟨a, b | aaaaa=b, bbbbb=1⟩

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Contents

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

Properties

Commutative structure

Finite structure

The order of x is the least n such that xn = 1.

Staircase diagram

Right Cayley graph

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. a25 ⇒ 1 [2]
2. b ⇒ a5 [1]
# ab:aaaaa=b,bbbbb=1 a/b - -
aaaaaaaaaaaaaaaaaaaaaaaaa=1
b=aaaaa

Same cardinality

7 unique, 10 total

Σ#PresentationPropertiesφ
104749⟨a, b | abba=b, baab=a⟩Noc 25
106381⟨a, b | aab=a, bbbbb=b⟩Noc 251
106536⟨a, b | aaa=a, aaba=bb⟩Noc 25
1114342⟨a, b | aaaa=a, aabba=b⟩Noc 25
1114869⟨a, b | abba=b, bbabb=a⟩Noc 252
1119634⟨a, b | aba=a, aaaaa=bb⟩Noc 25
1119706⟨a, b | aba=b, aaaaa=bb⟩Noc 25

Isomorphic instances

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

3 total

Σ#PresentationMapping
1113298⟨a, b | aaaaa=1, abbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaaaaaa, φ(b) = a
1113305⟨a, b | aaaaa=1, babbbb=1⟩φ(a) = aaaaaaaaaaaaaaaaaaaa, φ(b) = a
1113307⟨a, b | aaaaa=1, bbabbb=1⟩φ(a) = aaaaaaaaaaaaaaaaaaaa, φ(b) = a