#6381 ⟨a, b | aab=a, bbbbb=b⟩

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Contents

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

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.

Right Cayley graph

Left 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. a5 ⇒ a [7]
2. ab ⇒ a4 [6]
3. b5 ⇒ b [2]
# ab:aab=a,bbbbb=b a/b - -
aaaaa=a
ab=aaaa
bbbbb=b

Same cardinality

7 unique, 12 total

Σ#PresentationPropertiesDescriptionφ
104749⟨a, b | abba=b, baab=a⟩Noc 25
106536⟨a, b | aaa=a, aaba=bb⟩Noc 25
1111133⟨a, b | aaaaa=b, bbbbb=1⟩Grp Com 25ℤ253
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.

1 total

Σ#PresentationMapping
1114792⟨a, b | abab=a, bbbbb=b⟩φ(a) = aa, φ(b) = b