#1307 ⟨a, b | bab=aaa, bbb=1⟩

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Quick links

  1. Properties
  2. Elements
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. 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. a25 ⇒ a [8]
2. aba24 ⇒ ab [10]
3. a4b ⇒ ba4 [3]
4. b2a ⇒ aba22 [9]
5. bab ⇒ a3 [1]
6. ba2b ⇒ a3ba3 [5]
7. ab2 ⇒ a2ba21 [11]
8. aba3b ⇒ ba3ba7 [6]
9. b3 ⇒ 1 [2]
# ab:bab=aaa,bbb=1 a/b
aaaaaaaaaaaaaaaaaaaaaaaaa=a
abaaaaaaaaaaaaaaaaaaaaaaaa=ab
aaaab=baaaa
bba=abaaaaaaaaaaaaaaaaaaaaaa
bab=aaa
baab=aaabaaa
abb=aabaaaaaaaaaaaaaaaaaaaaa
abaaab=baaabaaaaaaa
bbb=1

Isomorphic instances

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

9 total

Σ#PresentationMapping
92441⟨a, b | aaa=1, abbba=b⟩φ(a) = bb, φ(b) = a
107780⟨a, b | aaa=1, aabbb=ba⟩φ(a) = b, φ(b) = a
108072⟨a, b | aaa=1, abbb=baa⟩φ(a) = bb, φ(b) = a
1121701⟨a, b | aaa=1, aabbbaa=b⟩φ(a) = b, φ(b) = a
1121723⟨a, b | aaa=1, abababa=b⟩φ(a) = bb, φ(b) = ab
1122243⟨a, b | aaa=1, aabbba=ab⟩φ(a) = bb, φ(b) = a
1122755⟨a, b | aaa=1, aabaa=bbb⟩φ(a) = bb, φ(b) = a
1122840⟨a, b | aaa=1, babab=aba⟩φ(a) = b, φ(b) = aaa
1123285⟨a, b | aaa=1, abbb=aaba⟩φ(a) = b, φ(b) = a