#15158 ⟨a, b | aab=ba, ababab=1⟩

Up:Monoid enumeration
Prev:#15137 ⟨a, b | aab=ba, aaaaaa=1⟩
Next:#15200 ⟨a, b | aab=ba, bbbbbb=1⟩

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.
The order of x is the least n such that xn = 1.

Cayley table

1aba2abb2a3a2bab2a4a3ba2b2a5a4ba3b2a6a5ba4b2a6ba5b2a6b2
11aba2abb2a3a2bab2a4a3ba2b2a5a4ba3b2a6a5ba4b2a6ba5b2a6b2
aaa2aba3a2bab2a4a3ba2b2a5a4ba3b2a6a5ba4b21a6ba5b2ba6b2b2
bba2bb2a4ba2b21a6ba4b2a2aba6b2a4a3bab2a6a5ba3b2aa5b2a3a5
a2a2a3a2ba4a3ba2b2a5a4ba3b2a6a5ba4b21a6ba5b2aba6b2abb2ab2
ababa3bab2a5ba3b2aba5b2a3a2bb2a5a4ba2b21a6ba4b2a2a6b2a4a6
b2b2a4b21ab2a4ba5b2aa4ba2b2a5aba6b2a2a5ba3b2a6a2ba3a6ba3b
a3a3a4a3ba5a4ba3b2a6a5ba4b21a6ba5b2aba6b2a2abb2a2bab2a2b2
a2ba2ba4ba2b2a6ba4b2a2aba6b2a4a3bab2a6a5ba3b2aba5b2a3b2a51
ab2ab2a5b2aa2b2a5aba6b2a2a5ba3b2a6a2bb2a3a6ba4b21a3ba4ba4b
a4a4a5a4ba6a5ba4b21a6ba5b2aba6b2a2abb2a3a2bab2a3ba2b2a3b2
a3ba3ba5ba3b2ba5b2a3a2bb2a5a4ba2b21a6ba4b2a2aba6b2a4ab2a6a
a2b2a2b2a6b2a2a3b2a6a2bb2a3a6ba4b21a3bab2a4ba5b2aa4ba5aba5b
a5a5a6a5b1a6ba5b2aba6b2a2abb2a3a2bab2a4a3ba2b2a4ba3b2a4b2
a4ba4ba6ba4b2aba6b2a4a3bab2a6a5ba3b2aba5b2a3a2bb2a5a2b21a2
a3b2a3b2b2a3a4b21a3bab2a4ba5b2aa4ba2b2a5aba6b2a2a5ba6a2ba6b
a6a61a6baba6b2a2abb2a3a2bab2a4a3ba2b2a5a4ba3b2a5ba4b2a5b2
a5ba5bba5b2a2bb2a5a4ba2b21a6ba4b2a2aba6b2a4a3bab2a6a3b2aa3
a4b2a4b2ab2a4a5b2aa4ba2b2a5aba6b2a2a5ba3b2a6a2bb2a3a6b1a3bb
a6ba6baba6b2a3bab2a6a5ba3b2aba5b2a3a2bb2a5a4ba2b21a4b2a2a4
a5b2a5b2a2b2a5a6b2a2a5ba3b2a6a2bb2a3a6ba4b21a3bab2a4baa4bab
a6b2a6b2a3b2a6b2a3a6ba4b21a3bab2a4ba5b2aa4ba2b2a5aba2a5ba2b

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. a7 ⇒ 1 [6]
2. ba ⇒ a2b [1]
3. b3 ⇒ 1 [5]
# ab:aab=ba,ababab=1 reversed:a/b - -
aaaaaaa=1
ba=aab
bbb=1

Same cardinality

8 unique, 63 total

Σ#PresentationPropertiesDescriptionφ
8402⟨a, b | aaa=ab, bbb=1⟩Noc 2139
91599⟨a, b | aaa=ab, bbb=b⟩Noc 211
105309⟨a, b | aaa=aa, bbb=ab⟩Noc 21
106265⟨a, b | aaa=a, abbbb=b⟩Noc 212
1111127⟨a, b | aaaaa=b, abbbb=1⟩Grp Com 21ℤ2113
1113091⟨a, b | bab=aaa, abba=b⟩Noc 21
1113248⟨a, b | bab=aaa, bbb=aa⟩Noc 21
1119212⟨a, b | aba=b, baaaab=a⟩Noc 21

Isomorphic instances

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

1 total

Σ#PresentationMapping
1115179⟨a, b | aab=ba, bababa=1⟩φ(a) = a, φ(b) = b