#16069 ⟨a, b | aaa=bb, abbb=ba⟩

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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. Anti-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.

1aba2abb2a2bab2b3a2b2ab3b4a2b3
11aba2abb2a2bab2b3a2b2ab3b4a2b3
aaa2abb2a2bab2b3a2b2ab3b4a2b3ab2b3
bbab3b2a2b3ab2b3a2b2ab3b4a2b3ab2b3a2b2
a2a2b2a2bab2b3a2b2ab3b4a2b3ab2b3a2b2ab3
ababa2b3ab2b3a2b2ab3b4a2b3ab2b3a2b2ab3b4
b2b2ab2b3a2b2ab3b4a2b3ab2b3a2b2ab3b4a2b3
a2ba2bb3a2b2ab3b4a2b3ab2b3a2b2ab3b4a2b3ab2
ab2ab2a2b2ab3b4a2b3ab2b3a2b2ab3b4a2b3ab2b3
b3b3ab3b4a2b3ab2b3a2b2ab3b4a2b3ab2b3a2b2
a2b2a2b2b4a2b3ab2b3a2b2ab3b4a2b3ab2b3a2b2ab3
ab3ab3a2b3ab2b3a2b2ab3b4a2b3ab2b3a2b2ab3b4
b4b4ab2b3a2b2ab3b4a2b3ab2b3a2b2ab3b4a2b3
a2b3a2b3b3a2b2ab3b4a2b3ab2b3a2b2ab3b4a2b3ab2

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. b5 ⇒ b3 [6]
2. ab4 ⇒ ab2 [7]
3. ba ⇒ ab3 [2]
4. a3 ⇒ b2 [1]
# ab:aaa=bb,abbb=ba b/a - -
bbbbb=bbb
abbbb=abb
ba=abbb
aaa=bb

Same cardinality

26 unique, 302 total

Σ#PresentationPropertiesDescriptionφ
91328⟨a, b | aaaa=b, abbb=1⟩Grp Com 13ℤ13189
92118⟨a, b | aba=b, baab=a⟩Noc 134
104642⟨a, b | aaaa=b, abbb=a⟩Noc Com 13ℕ(13 = 1)10
104643⟨a, b | aaaa=b, abbb=b⟩Noc Com 13ℕ(13 = 4)6
104683⟨a, b | aaab=b, abba=a⟩Noc 1316
105065⟨a, b | aaa=ab, babb=b⟩Noc 137
105334⟨a, b | aaa=ab, bba=bb⟩Noc 13
105336⟨a, b | aaa=ab, bbb=ab⟩Noc 131
105337⟨a, b | aaa=ab, bbb=ba⟩Noc 13
105340⟨a, b | aaa=bb, aab=ba⟩Noc 13
106305⟨a, b | aaa=b, abbbb=b⟩Noc Com 13ℕ(13 = 3)2
107143⟨a, b | bb=aa, aaab=ba⟩Noc 131
1112240⟨a, b | aaab=aa, bbbb=a⟩Noc Com 13ℕ(13 = 8)1
1112268⟨a, b | aaab=ab, bbbb=a⟩Noc Com 13ℕ(13 = 5)3
1112324⟨a, b | aaab=bb, bbbb=a⟩Noc Com 13ℕ(13 = 2)14
1114647⟨a, b | aaba=b, babbb=a⟩Noc Com 132
1115520⟨a, b | aaa=bb, aabbb=b⟩Noc Com 132
1116012⟨a, b | aaa=ab, abbb=bb⟩Noc 13
1116205⟨a, b | aab=ab, bbbb=aa⟩Noc 132
1116459⟨a, b | aba=bb, abbb=aa⟩Noc 131
1116506⟨a, b | aab=ab, bbb=aaa⟩Noc 131
1116515⟨a, b | aab=ba, bbb=aaa⟩Noc 13
1118958⟨a, b | aab=a, bbbbbb=a⟩Noc Com 13ℕ(13 = 6)4
1120046⟨a, b | aab=a, bbbb=bba⟩Noc 13
1120927⟨a, b | ab=aa, aaaa=bbb⟩Noc 137
1120991⟨a, b | ab=aa, baaa=bbb⟩Noc 133

Anti-isomorphic instances

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

1 total

Σ#PresentationMapping
1116075⟨a, b | aaa=bb, babb=ab⟩φ(a) = a, φ(b) = b