#7795 ⟨a, b | aaa=1, abbab=ba⟩

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Properties

Elements

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.

Complete rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
  1. a3 ⇒ 1
  2. b2ab ⇒ a2ba
  3. ab3 ⇒ (ba)3a
  4. (ab)2a2b ⇒ ba2b2a2
  5. (aba)2b ⇒ b(ba2)2
  6. ab(a2b)2 ⇒ bab2a
  7. a2b2a2b ⇒ ba2(ba)2
  8. a2bab2 ⇒ (ba2)3
  9. a(ab)3 ⇒ b3a
  10. (a2b)2b ⇒ b(aba)2
  11. (ba)2b2 ⇒ (aba)2
  12. ab2(a2b)2 ⇒ b2(a2b)2a
  13. abab2a2b ⇒ bab2a2ba
  14. b4a2b ⇒ aba2
  15. b3a2b2 ⇒ abab2a2
  16. (ab)5 ⇒ b5a2
  17. b7 ⇒ b 
# ab:aaa=1,abbab=ba a/b
aaa=1
bbab=aaba
abbb=bababaa
ababaab=baabbaa
abaabab=bbaabaa
abaabaab=babba
aabbaab=baababa
aababb=baabaabaa
aababab=bbba
aabaabb=babaaba
bababb=abaaba
abbaabaab=bbaabaaba
ababbaab=babbaaba
bbbbaab=abaa
bbbaabb=ababbaa
ababababab=bbbbbaa
bbbbbbb=b

Right Cayley graph

Left Cayley graph

Other isomorphic instances

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

1 total

Σ#PresentationMapping
1122259⟨a, b | aaa=1, abaabb=ba⟩φ(a) = a, φ(b) = ab

Other anti-isomorphic instances

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

2 total

Σ#PresentationMapping
1121691⟨a, b | aaa=1, aababba=b⟩φ(a) = a, φ(b) = b
1123308⟨a, b | aaa=1, babb=abaa⟩φ(a) = a, φ(b) = b