#5073 ⟨a, b | aaa=ab, bbbb=b⟩

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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

Properties

Finite structure

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
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. a9 ⇒ a3 [3]
2. ab ⇒ a3 [1]
3. b4 ⇒ b [2]
# ab:aaa=ab,bbbb=b a/b - -
aaaaaaaaa=aaa
ab=aaa
bbbb=b

Same cardinality

4 unique, 40 total

Σ#PresentationPropertiesφ
91345⟨a, b | aaab=a, bbbb=1⟩Noc 3613
92300⟨a, b | aaa=1, ababbb=1⟩Grp 3623
1116044⟨a, b | aaa=ab, bbbb=bb⟩Noc 36
1118959⟨a, b | aab=a, bbbbbb=b⟩Noc 36

Isomorphic instances

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

1 total

Σ#PresentationMapping
1112995⟨a, b | abb=aaa, bbbb=b⟩φ(a) = a, φ(b) = bb