#526 ⟨a, b | aab=a, bbbb=1⟩

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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.

1aba2bab2a3ba2b2ab3a4ba3b2a2b3aba4b2a3b3a2b2a4b3a3b3a4
11aba2bab2a3ba2b2ab3a4ba3b2a2b3aba4b2a3b3a2b2a4b3a3b3a4
aaa2a4a3aa3a4a2a4a2aa3aa3a4a2a4a3aa2
bbbab2ba2b2ab3ba3b2a2b3a1ba4b2a3b3a2ab2a4b3a3a2b3a4a3a4
a2a2a3aa4a2a4aa3aa3a2a4a2a4aa3aa4a2a3
bababa2ba4ba3baba3ba4ba2ba4ba2baba3baba3ba4ba2ba4ba3baba2
b2b2b2ab3b2a2b3a1b2a3b3a2abb2a4b3a3a2bab3a4a3ba2a4ba3ba4
a3a3a4a2aa3aa2a4a2a4a3aa3aa2a4a2aa3a4
ba2ba2ba3baba4ba2ba4baba3baba3ba2ba4ba2ba4baba3baba4ba2ba3
b2ab2ab2a2b2a4b2a3b2ab2a3b2a4b2a2b2a4b2a2b2ab2a3b2ab2a3b2a4b2a2b2a4b2a3b2ab2a2
b3b3b3a1b3a2abb3a3a2bab2b3a4a3ba2b2aa4ba3b2a2ba4b2a3b2a4
a4a4aa3a2a4a2a3aa3aa4a2a4a2a3aa3a2a4a
ba3ba3ba4ba2baba3baba2ba4ba2ba4ba3baba3baba2ba4ba2baba3ba4
b2a2b2a2b2a3b2ab2a4b2a2b2a4b2ab2a3b2ab2a3b2a2b2a4b2a2b2a4b2ab2a3b2ab2a4b2a2b2a3
b3ab3ab3a2b3a4b3a3b3ab3a3b3a4b3a2b3a4b3a2b3ab3a3b3ab3a3b3a4b3a2b3a4b3a3b3ab3a2
ba4ba4baba3ba2ba4ba2ba3baba3baba4ba2ba4ba2ba3baba3ba2ba4ba
b2a3b2a3b2a4b2a2b2ab2a3b2ab2a2b2a4b2a2b2a4b2a3b2ab2a3b2ab2a2b2a4b2a2b2ab2a3b2a4
b3a2b3a2b3a3b3ab3a4b3a2b3a4b3ab3a3b3ab3a3b3a2b3a4b3a2b3a4b3ab3a3b3ab3a4b3a2b3a3
b2a4b2a4b2ab2a3b2a2b2a4b2a2b2a3b2ab2a3b2ab2a4b2a2b2a4b2a2b2a3b2ab2a3b2a2b2a4b2a
b3a3b3a3b3a4b3a2b3ab3a3b3ab3a2b3a4b3a2b3a4b3a3b3ab3a3b3ab3a2b3a4b3a2b3ab3a3b3a4
b3a4b3a4b3ab3a3b3a2b3a4b3a2b3a3b3ab3a3b3ab3a4b3a2b3a4b3a2b3a3b3ab3a3b3a2b3a4b3a

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. a5 ⇒ a [6]
2. ab ⇒ a4 [5]
3. b4 ⇒ 1 [2]
# ab:aab=a,bbbb=1 a/b - -
aaaaa=a
ab=aaaa
bbbb=1

Same cardinality

20 unique, 46 total

Σ#PresentationPropertiesDescriptionφ
92207⟨a, b | ab=aa, bbbb=b⟩Noc 201
104095⟨a, b | baa=abb, abab=1⟩Grp 205
104212⟨a, b | aaaaa=1, abbbb=1⟩Grp Com 20ℤ2016
105349⟨a, b | aaa=bb, bab=aa⟩Noc 201
106728⟨a, b | aba=a, aaaa=bb⟩Noc 20
106764⟨a, b | aba=b, aaaa=bb⟩Noc 20
107117⟨a, b | ab=aa, bbbb=bb⟩Noc 20
1112159⟨a, b | aaaa=aa, abbb=b⟩Noc 20
1112322⟨a, b | aaab=bb, bbba=a⟩Noc 20
1114367⟨a, b | aaaa=a, bbbbb=a⟩Noc Com 20ℕ(20 = 5)
1114407⟨a, b | aaaa=b, bbbbb=a⟩Noc Com 20ℕ(20 = 1)
1114408⟨a, b | aaaa=b, bbbbb=b⟩Noc Com 20ℕ(20 = 4)
1115933⟨a, b | aba=bb, baaab=a⟩Noc 20
1116124⟨a, b | aab=aa, baba=bb⟩Noc 20
1116339⟨a, b | aba=aa, aaaa=bb⟩Noc 20
1116343⟨a, b | aba=aa, aaab=bb⟩Noc 20
1116545⟨a, b | aba=bb, abb=aaa⟩Noc 20
1118619⟨a, b | aba=b, aaaaabb=1⟩Grp 203
1119503⟨a, b | aab=a, bbbbb=bb⟩Noc 20
1121047⟨a, b | ab=aa, bbbb=bbb⟩Noc 20

Isomorphic instances

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

10 total

Σ#PresentationMapping
8617⟨a, b | ab=aa, bbbb=1⟩φ(a) = aaa, φ(b) = b
91412⟨a, b | abab=a, bbbb=1⟩φ(a) = aa, φ(b) = b
91565⟨a, b | aba=ab, bbbb=1⟩φ(a) = ba, φ(b) = b
103905⟨a, b | aabb=ab, bbbb=1⟩φ(a) = a, φ(b) = b
103967⟨a, b | abba=ab, bbbb=1⟩φ(a) = a, φ(b) = b
103978⟨a, b | abbb=aa, bbbb=1⟩φ(a) = a, φ(b) = b
104064⟨a, b | abb=aba, aaaa=1⟩φ(a) = b, φ(b) = aaa
1110987⟨a, b | abab=abb, bbbb=1⟩φ(a) = ba, φ(b) = b
1111064⟨a, b | abbb=aba, bbbb=1⟩φ(a) = aa, φ(b) = b
1117651⟨a, b | aaaa=1, ababa=ab⟩φ(a) = b, φ(b) = aa

Anti-isomorphic instances

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

9 total

Σ#PresentationMapping
103986⟨a, b | baab=ab, bbbb=1⟩φ(a) = a, φ(b) = b
104054⟨a, b | abb=aab, aaaa=1⟩φ(a) = bbb, φ(b) = a
105701⟨a, b | aaaa=1, aaabb=b⟩φ(a) = bbb, φ(b) = a
105711⟨a, b | aaaa=1, abaab=b⟩φ(a) = bbb, φ(b) = a
1111001⟨a, b | abab=bab, bbbb=1⟩φ(a) = ba, φ(b) = b
1117099⟨a, b | aaaa=1, aaabab=b⟩φ(a) = b, φ(b) = ba
1117119⟨a, b | aaaa=1, abaaab=b⟩φ(a) = b, φ(b) = ba
1117635⟨a, b | aaaa=1, aabab=ab⟩φ(a) = b, φ(b) = aa
1117669⟨a, b | aaaa=1, baaab=ab⟩φ(a) = b, φ(b) = aa