#12981 ⟨a, b | abb=aaa, baaa=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

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.

1aba2abb2ab2b3ab3b4ab4b5ab5b6ab6
11aba2abb2ab2b3ab3b4ab4b5ab5b6ab6
aaa2abab2ab5ab2ab6ab3abab4ab2ab5ab3ab6ab4
bbb5b2b3b6b3bb4b2b5b3b6b4bb5
a2a2ab2ab5ab6ab3ab6ab4abab5ab2ab6ab3abab4ab2
ababab5ab2ab3ab6ab3abab4ab2ab5ab3ab6ab4abab5
b2b2b6b3b4bb4b2b5b3b6b4bb5b2b6
ab2ab2ab6ab3ab4abab4ab2ab5ab3ab6ab4abab5ab2ab6
b3b3bb4b5b2b5b3b6b4bb5b2b6b3b
ab3ab3abab4ab5ab2ab5ab3ab6ab4abab5ab2ab6ab3ab
b4b4b2b5b6b3b6b4bb5b2b6b3bb4b2
ab4ab4ab2ab5ab6ab3ab6ab4abab5ab2ab6ab3abab4ab2
b5b5b3b6bb4bb5b2b6b3bb4b2b5b3
ab5ab5ab3ab6abab4abab5ab2ab6ab3abab4ab2ab5ab3
b6b6b4bb2b5b2b6b3bb4b2b5b3b6b4
ab6ab6ab4abab2ab5ab2ab6ab3abab4ab2ab5ab3ab6ab4

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. b7 ⇒ b [8]
2. ba ⇒ b5 [7]
3. a2b ⇒ ab5 [9]
4. a3 ⇒ ab2 [1]
# ab:abb=aaa,baaa=b b/a - -
bbbbbbb=b
ba=bbbbb
aab=abbbbb
aaa=abb

Same cardinality

22 unique, 187 total

Σ#PresentationPropertiesDescriptionφ
91602⟨a, b | aaa=bb, aba=a⟩Noc 152
91603⟨a, b | aaa=bb, aba=b⟩Noc 151
91751⟨a, b | aaab=1, bbbbb=1⟩Grp Com 15ℤ15138
91993⟨a, b | aaa=a, abbb=b⟩Noc 154
105342⟨a, b | aaa=bb, aba=aa⟩Noc 151
105344⟨a, b | aaa=bb, aba=bb⟩Noc 152
106276⟨a, b | aaa=a, bbbbb=a⟩Noc Com 15ℕ(15 = 5)
106316⟨a, b | aaa=b, bbbbb=a⟩Noc Com 15ℕ(15 = 1)
106317⟨a, b | aaa=b, bbbbb=b⟩Noc Com 15ℕ(15 = 3)
1112985⟨a, b | abb=aaa, baba=b⟩Noc 157
1113123⟨a, b | aab=aaa, aba=bb⟩Noc 15
1113191⟨a, b | abb=aaa, bbb=ab⟩Noc 151
1113192⟨a, b | abb=aaa, bbb=ba⟩Noc 15
1115439⟨a, b | aaa=aa, bbbbb=a⟩Noc Com 15ℕ(15 = 10)
1115503⟨a, b | aaa=ab, bbbbb=a⟩Noc Com 15ℕ(15 = 6)1
1115543⟨a, b | aaa=bb, bbbbb=a⟩Noc Com 15ℕ(15 = 2)1
1116041⟨a, b | aaa=ab, bbbb=aa⟩Noc 15
1116142⟨a, b | aab=aa, bbbb=ab⟩Noc 15
1119374⟨a, b | aaa=b, bbbbb=ab⟩Noc Com 15ℕ(15 = 4)
1119502⟨a, b | aab=a, bbbbb=ba⟩Noc 151
1120144⟨a, b | aab=b, bbaa=aaa⟩Noc 15
1120819⟨a, b | ab=aa, bbaaa=bb⟩Noc 156

Isomorphic instances

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

3 total

Σ#PresentationMapping
1112987⟨a, b | abb=aaa, babb=b⟩φ(a) = a, φ(b) = b
1112991⟨a, b | abb=aaa, bbab=b⟩φ(a) = a, φ(b) = b
1112993⟨a, b | abb=aaa, bbba=b⟩φ(a) = a, φ(b) = b