#1345 ⟨a, b | aaab=a, 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. a9 ⇒ a [6]
2. ab ⇒ a7 [5]
3. b4 ⇒ 1 [2]
# ab:aaab=a,bbbb=1 a/b - -
aaaaaaaaa=a
ab=aaaaaaa
bbbb=1

Same cardinality

4 unique, 28 total

Σ#PresentationPropertiesφ
92300⟨a, b | aaa=1, ababbb=1⟩Grp 3623
105073⟨a, b | aaa=ab, bbbb=b⟩Noc 361
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.

5 total

Σ#PresentationMapping
91465⟨a, b | aaa=ab, bbbb=1⟩φ(a) = ba, φ(b) = b
1110725⟨a, b | aaab=abb, bbbb=1⟩φ(a) = ba, φ(b) = b
1111062⟨a, b | abbb=aaa, bbbb=1⟩φ(a) = a, φ(b) = b
1117662⟨a, b | aaaa=1, abbba=ab⟩φ(a) = b, φ(b) = a
1117673⟨a, b | aaaa=1, baabb=ba⟩φ(a) = b, φ(b) = a

Anti-isomorphic instances

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

8 total

Σ#PresentationMapping
1110753⟨a, b | aaab=bab, bbbb=1⟩φ(a) = ba, φ(b) = b
1117103⟨a, b | aaaa=1, aaabbb=b⟩φ(a) = b, φ(b) = ba
1117123⟨a, b | aaaa=1, abaabb=b⟩φ(a) = b, φ(b) = ba
1117125⟨a, b | aaaa=1, ababab=b⟩φ(a) = b, φ(b) = ba
1117131⟨a, b | aaaa=1, abbaab=b⟩φ(a) = b, φ(b) = ba
1117643⟨a, b | aaaa=1, aabbb=ab⟩φ(a) = b, φ(b) = a
1117672⟨a, b | aaaa=1, baabb=ab⟩φ(a) = b, φ(b) = a
1117676⟨a, b | aaaa=1, babab=ab⟩φ(a) = b, φ(b) = a