#1434 ⟨a, b | baab=a, bbbb=1⟩

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Quick links

  1. Properties
  2. Elements
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality
  7. 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. a13 ⇒ a [13]
2. aba12 ⇒ ab [12]
3. a3b ⇒ ba3 [3]
4. ba2b ⇒ a [1]
5. ab2 ⇒ baba2 [7]
6. a(ab)2 ⇒ b2a [6]
7. b3a ⇒ a2b [5]
8. b(ab)2 ⇒ aba10 [10]
9. b4 ⇒ 1 [2]
# 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

Same cardinality

1 unique, 3 total

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

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