#7529 ⟨a, b | aaa=1, abbbba=b⟩

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Contents

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

Properties

Finite structure

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
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. b91 ⇒ b [18]
2. bab90 ⇒ ba [19]
3. b5a ⇒ ab5 [5]
4. a2b ⇒ b4a [3]
5. ab3a ⇒ bab86 [20]
6. ab4a ⇒ b [2]
7. ba2 ⇒ ab4 [4]
8. (ba)2 ⇒ ab2ab9 [9]
9. b4ab2a ⇒ abab86 [21]
10. a3 ⇒ 1 [1]
11. a(b2a)2 ⇒ bab2ab76 [22]
# ab:aaa=1,abbbba=b b/a - -
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
babbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=ba
bbbbba=abbbbb
aab=bbbba
abbba=babbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
abbbba=b
baa=abbbb
baba=abbabbbbbbbbb
bbbbabba=ababbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
aaa=1
abbabba=babbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

Isomorphic instances

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

4 total

Σ#PresentationMapping
108092⟨a, b | aaa=1, bbbb=aba⟩φ(a) = aa, φ(b) = b
1122248⟨a, b | aaa=1, aabbbb=ba⟩φ(a) = aa, φ(b) = b
1122820⟨a, b | aaa=1, abbbb=baa⟩φ(a) = a, φ(b) = b
1122854⟨a, b | aaa=1, bbabb=aba⟩φ(a) = aa, φ(b) = bb