#2584 ⟨a, b | aaa=1, abbb=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. b27 ⇒ b
  2. ba ⇒ ab3
  3. a3 ⇒ 1 
# ab:aaa=1,abbb=ba b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbb=b
ba=abbb
aaa=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.

6 total

Σ#PresentationMapping
107507⟨a, b | aaa=1, aabbba=b⟩φ(a) = aa, φ(b) = b
108045⟨a, b | aaa=1, aaba=bbb⟩φ(a) = a, φ(b) = b
1122263⟨a, b | aaa=1, ababab=ba⟩φ(a) = a, φ(b) = aab
1122774⟨a, b | aaa=1, aabbb=aba⟩φ(a) = a, φ(b) = b
1122776⟨a, b | aaa=1, aabbb=baa⟩φ(a) = aa, φ(b) = b
1122811⟨a, b | aaa=1, abbba=aab⟩φ(a) = aa, φ(b) = b