#1523 ⟨a, b | aab=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. Same cardinality
  7. Isomorphic instances
  8. 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. a16 ⇒ a [3]
2. ba ⇒ a2b [1]
3. b4 ⇒ 1 [2]
# ab:aab=ba,bbbb=1 reversed:a/b - -
aaaaaaaaaaaaaaaa=a
ba=aab
bbbb=1

Same cardinality

2 unique, 9 total

Σ#PresentationPropertiesφ
105442⟨a, b | aaaa=1, ababbb=1⟩Grp 642
1010273⟨a, b | aa=1, ababa=bbb⟩Noc 645

Isomorphic instances

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

4 total

Σ#PresentationMapping
103947⟨a, b | abab=ba, bbbb=1⟩φ(a) = abbb, φ(b) = b
1110939⟨a, b | aabb=bab, bbbb=1⟩φ(a) = a, φ(b) = b
1117101⟨a, b | aaaa=1, aaabba=b⟩φ(a) = b, φ(b) = a
1117648⟨a, b | aaaa=1, abaab=ba⟩φ(a) = bbb, φ(b) = abb

Anti-isomorphic instances

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

2 total

Σ#PresentationMapping
1111079⟨a, b | baab=abb, bbbb=1⟩φ(a) = a, φ(b) = b
1117626⟨a, b | aaaa=1, aaaba=bb⟩φ(a) = b, φ(b) = a