#14981 ⟨a, b | aaa=bb, aabaab=1⟩

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

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. a21 ⇒ 1 [6]
2. ab ⇒ ba8 [5]
3. b2 ⇒ a3 [1]
# ab:aaa=bb,aabaab=1 a/b - -
aaaaaaaaaaaaaaaaaaaaa=1
ab=baaaaaaaa
bb=aaa

Same cardinality

2 unique, 5 total

Σ#PresentationPropertiesφ
106101⟨a, b | aab=a, bbbbbb=1⟩Noc 423
1120647⟨a, b | ab=aa, bbbbbb=b⟩Noc 42

Isomorphic instances

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

7 total

Σ#PresentationMapping
1114989⟨a, b | aaa=bb, abaaba=1⟩φ(a) = a, φ(b) = b
1126093⟨a, b | aa=1, abbabbbbb=1⟩φ(a) = baa, φ(b) = a
1126108⟨a, b | aa=1, abbbbbabb=1⟩φ(a) = baa, φ(b) = a
1126158⟨a, b | aa=1, babbabbbb=1⟩φ(a) = baa, φ(b) = a
1126162⟨a, b | aa=1, babbbbbab=1⟩φ(a) = baa, φ(b) = a
1126175⟨a, b | aa=1, bbabbabbb=1⟩φ(a) = baa, φ(b) = a
1126704⟨a, b | aa=1, bbabbbbb=a⟩φ(a) = baa, φ(b) = a