#3914 ⟨a, b | aabb=ba, bbbb=1⟩

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Contents

  1. Properties
  2. Finite structure
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Isomorphic instances
  7. Anti-isomorphic instances

Properties

Finite structure

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:
Enter a word above to compute its normal form. Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Strategy:
Result: 1 
1
#RuleProof
1. a5 ⇒ a [19]
2. a4ba ⇒ ba [21]
3. b2a ⇒ a2ba2 [4]
4. (ba)2 ⇒ aba4 [15]
5. ba2ba ⇒ a3ba3 [25]
6. ba3ba ⇒ a2 [22]
7. ab2 ⇒ a3ba [24]
8. b4 ⇒ 1 [2]
# ab:aabb=ba,bbbb=1 a/b - -
aaaaa=a
aaaaba=ba
bba=aabaa
baba=abaaaa
baaba=aaabaaa
baaaba=aa
abb=aaaba
bbbb=1

Isomorphic instances

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

3 total

Σ#PresentationMapping
103992⟨a, b | babb=aa, bbbb=1⟩φ(a) = a, φ(b) = b
104090⟨a, b | baa=abb, aaaa=1⟩φ(a) = bbb, φ(b) = a
1111090⟨a, b | babb=aba, bbbb=1⟩φ(a) = aa, φ(b) = b

Anti-isomorphic instances

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

4 total

Σ#PresentationMapping
105707⟨a, b | aaaa=1, aabba=b⟩φ(a) = bbb, φ(b) = a
1111005⟨a, b | abab=bba, bbbb=1⟩φ(a) = ab, φ(b) = bbb
1117107⟨a, b | aaaa=1, aababa=b⟩φ(a) = bbb, φ(b) = ab
1117636⟨a, b | aaaa=1, aabab=ba⟩φ(a) = b, φ(b) = aa