#3387 ⟨a, b | aa=1, abbbbb=b⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality
  8. Isomorphic instances
  9. 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.

Cayley table

Idempotents are shown in bold.

1ababbab2abaab2b2ab3ab2aab3b3ab4ab3aab4b4aab4a
11ababbab2abaab2b2ab3ab2aab3b3ab4ab3aab4b4aab4a
aa1abbabaab2bab2ab2aab3b2ab3ab3aab4b3ab4ab4ab4a
bbbab2ab2b2ab3ab2aab3b3ab4ab3aab4b4aabab4abababa
abababaab2b2ab2aab3b2ab3ab3aab4b3ab4ab4abb4aabbaaba
bababab2b2ab2aab3b2ab3ab3aab4b3ab4ab4abb4aabbaaba
b2b2b2ab3ab3b3ab4ab3aab4b4aabab4ababaab2bab2ab2ab2a
abaabaabb2ab2b2ab3ab2aab3b3ab4ab3aab4b4aabab4abababa
ab2ab2ab2aab3b3ab3aab4b3ab4ab4abb4aabbab2abaab2b2aab2a
b2ab2ab2ab3b3ab3aab4b3ab4ab4abb4aabbab2abaab2b2aab2a
b3b3b3ab4ab4b4aabab4ababaab2bab2ab2aab3b2ab3ab3ab3a
ab2aab2aab2b3ab3b3ab4ab3aab4b4aabab4ababaab2bab2ab2ab2a
ab3ab3ab3aab4b4ab4abb4aabbab2abaab2b2ab3ab2aab3b3aab3a
b3ab3ab3ab4b4ab4abb4aabbab2abaab2b2ab3ab2aab3b3aab3a
b4b4b4aabbabaab2bab2ab2aab3b2ab3ab3aab4b3ab4ab4ab4a
ab3aab3aab3b4ab4b4aabab4ababaab2bab2ab2aab3b2ab3ab3ab3a
ab4ab4ab4ababbab2abaab2b2ab3ab2aab3b3ab4ab3aab4b4aab4a
b4ab4ab4babbab2abaab2b2ab3ab2aab3b3ab4ab3aab4b4aab4a
ab4aab4aab4abbabaab2bab2ab2aab3b2ab3ab3aab4b3ab4ab4ab4a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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. a2 ⇒ 1 [1]
2. bab ⇒ ab2 [4]
3. b5 ⇒ ab [3]
# ab:aa=1,abbbbb=b ab - -
aa=1
bab=abb
bbbbb=ab

Same cardinality

17 unique, 73 total

Σ#PresentationPropertiesDescriptionφ
81121⟨a, b | aa=1, abbba=b⟩Noc 1817
105521⟨a, b | aaab=1, bbbbbb=1⟩Grp Com 18ℤ1833
106732⟨a, b | aba=a, aaab=bb⟩Noc 18
106788⟨a, b | aba=b, baab=aa⟩Noc 18
106795⟨a, b | aba=b, bbbb=aa⟩Noc 18
107039⟨a, b | bb=aa, aaaba=b⟩Noc 182
108910⟨a, b | aa=a, bbbbb=ab⟩Noc 181
1112187⟨a, b | aaaa=ab, babb=b⟩Noc 182
1115797⟨a, b | aab=bb, bbbba=a⟩Noc 18
1116042⟨a, b | aaa=ab, bbbb=ab⟩Noc 18
1116043⟨a, b | aaa=ab, bbbb=ba⟩Noc 18
1116313⟨a, b | aab=bb, baba=aa⟩Noc 18
1118758⟨a, b | aaa=a, bbbbbb=a⟩Noc Com 18ℕ(18 = 6)
1118830⟨a, b | aaa=b, bbbbbb=a⟩Noc Com 18ℕ(18 = 1)
1118831⟨a, b | aaa=b, bbbbbb=b⟩Noc Com 18ℕ(18 = 3)
1119624⟨a, b | aab=b, bbbba=aa⟩Noc 18
1120914⟨a, b | bb=aa, ababa=aa⟩Noc 181

Isomorphic instances

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

22 total

Σ#PresentationMapping
93542⟨a, b | aa=1, bbbbb=ab⟩φ(a) = a, φ(b) = b
1126501⟨a, b | aa=1, aaabbbbb=b⟩φ(a) = a, φ(b) = b
1126579⟨a, b | aa=1, abaabbbb=b⟩φ(a) = a, φ(b) = b
1126589⟨a, b | aa=1, abababbb=b⟩φ(a) = a, φ(b) = b
1126595⟨a, b | aa=1, ababbabb=b⟩φ(a) = a, φ(b) = b
1126597⟨a, b | aa=1, ababbbab=b⟩φ(a) = a, φ(b) = b
1126611⟨a, b | aa=1, abbaabbb=b⟩φ(a) = a, φ(b) = b
1126615⟨a, b | aa=1, abbababb=b⟩φ(a) = a, φ(b) = b
1126617⟨a, b | aa=1, abbabbab=b⟩φ(a) = a, φ(b) = b
1126625⟨a, b | aa=1, abbbaabb=b⟩φ(a) = a, φ(b) = b
1126627⟨a, b | aa=1, abbbabab=b⟩φ(a) = a, φ(b) = b
1126631⟨a, b | aa=1, abbbbaab=b⟩φ(a) = a, φ(b) = b
1127081⟨a, b | aa=1, aabbbbb=ab⟩φ(a) = a, φ(b) = b
1127195⟨a, b | aa=1, baabbbb=ab⟩φ(a) = a, φ(b) = b
1127206⟨a, b | aa=1, bababbb=ab⟩φ(a) = a, φ(b) = b
1127210⟨a, b | aa=1, babbabb=ab⟩φ(a) = a, φ(b) = b
1127214⟨a, b | aa=1, babbbab=ab⟩φ(a) = a, φ(b) = b
1127224⟨a, b | aa=1, bbaabbb=ab⟩φ(a) = a, φ(b) = b
1127228⟨a, b | aa=1, bbababb=ab⟩φ(a) = a, φ(b) = b
1127689⟨a, b | aa=1, abbbbb=aab⟩φ(a) = a, φ(b) = b
1127692⟨a, b | aa=1, abbbbb=baa⟩φ(a) = a, φ(b) = b
1128287⟨a, b | aa=1, bbbbb=aaab⟩φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

9 total

Σ#PresentationMapping
1126553⟨a, b | aa=1, aabbbbba=b⟩φ(a) = a, φ(b) = b
1127082⟨a, b | aa=1, aabbbbb=ba⟩φ(a) = a, φ(b) = b
1127162⟨a, b | aa=1, abbbbba=ab⟩φ(a) = a, φ(b) = b
1127196⟨a, b | aa=1, baabbbb=ba⟩φ(a) = a, φ(b) = b
1127207⟨a, b | aa=1, bababbb=ba⟩φ(a) = a, φ(b) = b
1127211⟨a, b | aa=1, babbabb=ba⟩φ(a) = a, φ(b) = b
1127225⟨a, b | aa=1, bbaabbb=ba⟩φ(a) = a, φ(b) = b
1127690⟨a, b | aa=1, abbbbb=aba⟩φ(a) = a, φ(b) = b
1128288⟨a, b | aa=1, bbbbb=aaba⟩φ(a) = a, φ(b) = b