#1385 ⟨a, b | aaba=b, 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.
The order of x is the least n such that xn = 1.

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. a15 ⇒ 1 [15]
2. ab ⇒ ba7 [13]
3. b4 ⇒ 1 [2]
# ab:aaba=b,bbbb=1 a/b - -
aaaaaaaaaaaaaaa=1
ab=baaaaaaa
bbbb=1

Same cardinality

1 unique, 4 total

Σ#PresentationPropertiesφ
1111604⟨a, b | baabb=a, bbbbb=1⟩Noc 603

Isomorphic instances

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

28 total

Σ#PresentationMapping
103884⟨a, b | aaba=bb, bbbb=1⟩φ(a) = bbba, φ(b) = b
105432⟨a, b | aaaa=1, aababb=1⟩φ(a) = bbb, φ(b) = ba
105434⟨a, b | aaaa=1, aabbab=1⟩φ(a) = b, φ(b) = bbba
105439⟨a, b | aaaa=1, abaabb=1⟩φ(a) = b, φ(b) = bbba
105441⟨a, b | aaaa=1, ababba=1⟩φ(a) = bbb, φ(b) = ba
105443⟨a, b | aaaa=1, abbaab=1⟩φ(a) = bbb, φ(b) = ba
105450⟨a, b | aaaa=1, baabab=1⟩φ(a) = bbb, φ(b) = ba
105724⟨a, b | aaaa=1, baabb=a⟩φ(a) = bbb, φ(b) = bbba
1114092⟨a, b | aaba=b, abbabb=1⟩φ(a) = a, φ(b) = ba
1114110⟨a, b | aaba=b, babbab=1⟩φ(a) = a, φ(b) = ba
1114119⟨a, b | aaba=b, bbabba=1⟩φ(a) = a, φ(b) = ba
1116573⟨a, b | aaaa=1, aaabbab=1⟩φ(a) = b, φ(b) = a
1116584⟨a, b | aaaa=1, aabbaba=1⟩φ(a) = b, φ(b) = a
1116592⟨a, b | aaaa=1, abaaabb=1⟩φ(a) = b, φ(b) = a
1117144⟨a, b | aaaa=1, baabab=a⟩φ(a) = bbb, φ(b) = bba
1117657⟨a, b | aaaa=1, abbab=aa⟩φ(a) = b, φ(b) = a
1126067⟨a, b | aa=1, ababbabbb=1⟩φ(a) = bb, φ(b) = ba
1126070⟨a, b | aa=1, ababbbabb=1⟩φ(a) = bb, φ(b) = bbba
1126088⟨a, b | aa=1, abbababbb=1⟩φ(a) = bb, φ(b) = bbba
1126091⟨a, b | aa=1, abbabbbab=1⟩φ(a) = bb, φ(b) = ba
1126099⟨a, b | aa=1, abbbababb=1⟩φ(a) = bb, φ(b) = ba
1126100⟨a, b | aa=1, abbbabbab=1⟩φ(a) = bb, φ(b) = bbba
1126151⟨a, b | aa=1, bababbabb=1⟩φ(a) = bb, φ(b) = ba
1126152⟨a, b | aa=1, bababbbab=1⟩φ(a) = bb, φ(b) = bbba
1126156⟨a, b | aa=1, babbababb=1⟩φ(a) = bb, φ(b) = bbba
1126684⟨a, b | aa=1, babbabbb=a⟩φ(a) = bb, φ(b) = ba
1126686⟨a, b | aa=1, babbbabb=a⟩φ(a) = bb, φ(b) = bbba
1126700⟨a, b | aa=1, bbababbb=a⟩φ(a) = bb, φ(b) = bbba

Anti-isomorphic instances

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

5 total

Σ#PresentationMapping
1116571⟨a, b | aaaa=1, aaababb=1⟩φ(a) = b, φ(b) = a
1116581⟨a, b | aaaa=1, aababba=1⟩φ(a) = b, φ(b) = a
1116602⟨a, b | aaaa=1, abbaaab=1⟩φ(a) = b, φ(b) = a
1116614⟨a, b | aaaa=1, baaabab=1⟩φ(a) = b, φ(b) = a
1117653⟨a, b | aaaa=1, ababb=aa⟩φ(a) = b, φ(b) = a