#7795 ⟨a, b | aaa=1, abbab=ba⟩

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Quick links

  1. Properties
  2. Elements
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Isomorphic instances
  7. Anti-isomorphic instances

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.

Right Cayley graph

Left Cayley graph

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
#RuleProof
1. a3 ⇒ 1 [1]
2. b2ab ⇒ a2ba [3]
3. ab3 ⇒ (ba)3a [6]
4. (ab)2a2b ⇒ ba2b2a2 [15]
5. (aba)2b ⇒ b(ba2)2 [14]
6. ab(a2b)2 ⇒ bab2a [13]
7. a2b2a2b ⇒ ba2(ba)2 [16]
8. a2bab2 ⇒ (ba2)3 [8]
9. a(ab)3 ⇒ b3a [5]
10. (a2b)2b ⇒ b(aba)2 [11]
11. (ba)2b2 ⇒ (aba)2 [7]
12. ab2(a2b)2 ⇒ b2(a2b)2a [24]
13. abab2a2b ⇒ bab2a2ba [22]
14. b4a2b ⇒ aba2 [19]
15. b3a2b2 ⇒ abab2a2 [17]
16. (ab)5 ⇒ b5a2 [23]
17. b7 ⇒ b [20]
# 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

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

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