#3123 ⟨a, b | ab=a, bbaa=bb⟩

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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

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.

Cayley table

Idempotents are shown in bold.

1aba2bab2ba2b2a
11aba2bab2ba2b2a
aaa2aaa2aaa2
bbbab2ba2b2ab2b2b2a
a2a2aa2a2aa2a2a
bababa2bababa2bababa2
b2b2b2ab2b2b2ab2b2b2a
ba2ba2baba2ba2baba2ba2ba
b2ab2ab2b2ab2ab2b2ab2ab2

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. ab ⇒ a [1]
2. a3 ⇒ a [3]
3. b3 ⇒ b2 [4]
4. b2a2 ⇒ b2 [2]
# ab:ab=a,bbaa=bb ab - -
ab=a
aaa=a
bbb=bb
bbaa=bb

Same cardinality

33 unique, 1201 total

Σ#PresentationPropertiesDescriptionφ
7188⟨a, b | aab=1, bbbb=1⟩Grp Com 8ℤ8727
7330⟨a, b | aa=1, abba=b⟩Noc 860
8556⟨a, b | aba=b, aabb=1⟩Grp 828
8898⟨a, b | aa=a, bbbb=a⟩Noc Com 8ℕ(8 = 4)48
8918⟨a, b | aa=b, bbbb=a⟩Noc Com 8ℕ(8 = 1)34
8919⟨a, b | aa=b, bbbb=b⟩Noc Com 8ℕ(8 = 2)46
8961⟨a, b | aa=a, aba=bb⟩Noc 85
91605⟨a, b | aaa=bb, abb=b⟩Noc Com 8ℕ(8 = 3)55
91606⟨a, b | aaa=bb, bab=a⟩Noc Com 814
91615⟨a, b | aab=aa, baa=b⟩Noc 814
91650⟨a, b | aab=bb, baa=a⟩Noc 815
92206⟨a, b | ab=aa, bbbb=a⟩Noc Com 8ℕ(8 = 5)32
92220⟨a, b | bb=aa, aaab=a⟩Noc Com 822
92247⟨a, b | ab=aa, baa=bb⟩Noc 81
92256⟨a, b | ab=aa, bbb=aa⟩Noc 81
92258⟨a, b | ab=aa, bbb=ba⟩Noc 8
92883⟨a, b | aa=a, abbbb=b⟩Noc 814
93107⟨a, b | ab=a, baaa=bb⟩Noc 89
105191⟨a, b | aab=bb, aaaa=b⟩Noc Com 8ℕ(8 = 6)18
106664⟨a, b | aab=b, aaaa=ba⟩Noc 81
107057⟨a, b | ab=aa, aaaa=bb⟩Noc 87
109380⟨a, b | ab=a, bbbb=baa⟩Noc 82
1112606⟨a, b | abbb=aa, bbbb=a⟩Noc Com 8ℕ(8 = 7)14
1120047⟨a, b | aab=a, bbbb=bbb⟩Noc 8
1120051⟨a, b | aab=b, aaaa=abb⟩Noc Com 81
1124863⟨a, b | ab=a, aaaaaa=bb⟩Noc Com 8
1125112⟨a, b | ab=a, bbbbbb=aa⟩Noc Com 8
1125114⟨a, b | ab=a, bbbbbb=ba⟩Noc 8
1125411⟨a, b | ab=a, aaaaa=bbb⟩Noc Com 8
1125652⟨a, b | ab=a, bbbbb=aaa⟩Noc Com 8
1125658⟨a, b | ab=a, bbbbb=bba⟩Noc 8
1125897⟨a, b | ab=a, bbbb=aaaa⟩Noc Com 8
1125911⟨a, b | ab=a, bbbb=bbba⟩Noc 8

Isomorphic instances

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

10 total

Σ#PresentationMapping
109087⟨a, b | ab=a, bbaab=bb⟩φ(a) = a, φ(b) = b
109091⟨a, b | ab=a, bbaba=bb⟩φ(a) = a, φ(b) = b
109099⟨a, b | ab=a, bbbaa=bb⟩φ(a) = a, φ(b) = b
1112451⟨a, b | aabb=aa, bbba=b⟩φ(a) = b, φ(b) = a
1125067⟨a, b | ab=a, bbaabb=bb⟩φ(a) = a, φ(b) = b
1125075⟨a, b | ab=a, bbabab=bb⟩φ(a) = a, φ(b) = b
1125079⟨a, b | ab=a, bbabba=bb⟩φ(a) = a, φ(b) = b
1125091⟨a, b | ab=a, bbbaab=bb⟩φ(a) = a, φ(b) = b
1125095⟨a, b | ab=a, bbbaba=bb⟩φ(a) = a, φ(b) = b
1125103⟨a, b | ab=a, bbbbaa=bb⟩φ(a) = a, φ(b) = b