#16506 ⟨a, b | aab=ab, bbb=aaa⟩

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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.

1aba2bab2a3ba2b2aa4ba3b2a2b2a3
11aba2bab2a3ba2b2aa4ba3b2a2b2a3
aaa2ba3a3ba3b2a3a4ba3b2a3a4ba3b2a3b2a3
bbbab2ba2b2aa3ba3b2a2a4ba3b2a3a4a4
a2a2a3ba3a4ba3b2a3a4ba3b2a3a4ba3b2a3b2a3
bababa2b2a3ba3b2a3a4ba3b2a3a4ba3b2a3a4a4
b2b2b2aa3b2a2a4ba3b2a3a4ba3b2a3a4ba3ba3
a3a3a4ba3a4ba3b2a3a4ba3b2a3a4ba3b2a3b2a3
ba2ba2ba3b2a3ba3b2a3a4ba3b2a3a4ba3b2a3a4a4
b2ab2ab2a2a4b2a3a4ba3b2a3a4ba3b2a3a4ba3ba3
a4a4a4ba3a4ba3b2a3a4ba3b2a3a4ba3b2a3b2a3
ba3ba3ba3b2a3ba3b2a3a4ba3b2a3a4ba3b2a3a4a4
b2a2b2a2b2a3a4b2a3a4ba3b2a3a4ba3b2a3a4ba3ba3
b2a3b2a3b2a3a4b2a3a4ba3b2a3a4ba3b2a3a4ba3ba3

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 ⇒ a4 [3]
2. ba4 ⇒ ba3 [5]
3. ab ⇒ ba3 [4]
4. b3 ⇒ a3 [2]
# ab:aab=ab,bbb=aaa a/b - -
aaaaa=aaaa
baaaa=baaa
ab=baaa
bbb=aaa

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
1116069⟨a, b | aaa=bb, abbb=ba⟩Noc 131
1116205⟨a, b | aab=ab, bbbb=aa⟩Noc 132
1116459⟨a, b | aba=bb, abbb=aa⟩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

Isomorphic instances

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

1 total

Σ#PresentationMapping
1116542⟨a, b | aba=ab, bbb=aaa⟩φ(a) = a, φ(b) = b