#1307 ⟨a, b | bab=aaa, bbb=1⟩

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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. a25 ⇒ a
  2. aba24 ⇒ ab
  3. a4b ⇒ ba4
  4. b2a ⇒ aba22
  5. bab ⇒ a3
  6. ba2b ⇒ a3ba3
  7. ab2 ⇒ a2ba21
  8. aba3b ⇒ ba3ba7
  9. b3 ⇒ 1 
# 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

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.

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