#4594 ⟨a, b | abba=b, aaaaa=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. b31 ⇒ b
  2. bab30 ⇒ ba
  3. b3a ⇒ ab3
  4. ab2a ⇒ b
  5. ba2b30 ⇒ ba2
  6. a3b ⇒ b(ba)2
  7. a(ba)2 ⇒ ba2b28
  8. ba3 ⇒ abab2
  9. (ba2)2 ⇒ (aba)2b5
  10. b2(aba)2 ⇒ a2ba2b28
  11. a5 ⇒ 1 
# ab:abba=b,aaaaa=1 b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
babbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=ba
bbba=abbb
abba=b
baabbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=baa
aaab=bbaba
ababa=baabbbbbbbbbbbbbbbbbbbbbbbbbbbb
baaa=ababb
baabaa=abaababbbbb
bbabaaba=aabaabbbbbbbbbbbbbbbbbbbbbbbbbbbb
aaaaa=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.

3 total

Σ#PresentationMapping
105002⟨a, b | aba=bb, aaaaa=1⟩φ(a) = aaaa, φ(b) = b
1111612⟨a, b | babab=a, bbbbb=1⟩φ(a) = abb, φ(b) = a
1112860⟨a, b | bab=aba, aaaaa=1⟩φ(a) = aaaa, φ(b) = ab