#1434 ⟨a, b | baab=a, bbbb=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. a13 ⇒ a
  2. aba12 ⇒ ab
  3. a3b ⇒ ba3
  4. ba2b ⇒ a
  5. ab2 ⇒ baba2
  6. a(ab)2 ⇒ b2a
  7. b3a ⇒ a2b
  8. b(ab)2 ⇒ aba10
  9. b4 ⇒ 1 
# ab:baab=a,bbbb=1 a/b
aaaaaaaaaaaaa=a
abaaaaaaaaaaaa=ab
aaab=baaa
baab=a
abb=babaa
aabab=bba
bbba=aab
babab=abaaaaaaaaaa
bbbb=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 3 total

Σ#PresentationDescriptionRelated
1113293⟨a, b | aaaaa=1, ababbb=1⟩Finite non-Abelian group with 100 elements2 iso

Other isomorphic instances

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

9 total

Σ#PresentationMapping
91566⟨a, b | aba=bb, aaaa=1⟩φ(a) = bbb, φ(b) = a
104108⟨a, b | bab=aba, aaaa=1⟩φ(a) = bbb, φ(b) = ab
105713⟨a, b | aaaa=1, ababa=b⟩φ(a) = b, φ(b) = aab
1111051⟨a, b | abba=bab, bbbb=1⟩φ(a) = ababa, φ(b) = b
1111066⟨a, b | abbb=baa, bbbb=1⟩φ(a) = a, φ(b) = b
1111089⟨a, b | babb=aab, bbbb=1⟩φ(a) = a, φ(b) = bbb
1117121⟨a, b | aaaa=1, abaaba=b⟩φ(a) = b, φ(b) = aa
1117629⟨a, b | aaaa=1, aaabb=ba⟩φ(a) = bbb, φ(b) = a
1117639⟨a, b | aaaa=1, aabba=ab⟩φ(a) = b, φ(b) = a