#2118 ⟨a, b | aba=b, baab=a⟩

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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.

1aba2baa3ba2a4ba3a5ba4a6ba5
11aba2baa3ba2a4ba3a5ba4a6ba5
aaa2ba5a3ba4baa5ba2a6ba3aba4
bbbaa3ba2a4ba3a5ba4a6ba5aba2
a2a2a3ba4a4ba5a5ba6baaba2a2ba3
bababa2a2ba3a3ba4a4ba5a5ba6baa
a3a3a4ba3a5ba4a6ba5aba2baa3ba2
ba2ba2ba3aba4a2ba5a3ba4baa5ba2a6
a4a4a5ba2a6ba3aba4a2ba5a3ba4ba
ba3ba3ba4a6ba5aba2baa3ba2a4ba3a5
a5a5a6baaba2a2ba3a3ba4a4ba5a5b
ba4ba4ba5a5ba6baaba2a2ba3a3ba4a4
a6a6aba2baa3ba2a4ba3a5ba4a6ba5
ba5ba5ba4baa5ba2a6ba3aba4a2ba5a3

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. a7 ⇒ a [7]
2. ba6 ⇒ b [6]
3. ab ⇒ ba5 [5]
4. b2 ⇒ a3 [4]
# ab:aba=b,baab=a a/b - -
aaaaaaa=a
baaaaaa=b
ab=baaaaa
bb=aaa

Same cardinality

26 unique, 299 total

Σ#PresentationPropertiesDescriptionφ
91328⟨a, b | aaaa=b, abbb=1⟩Grp Com 13ℤ13189
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
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

Isomorphic instances

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

4 total

Σ#PresentationMapping
1119166⟨a, b | aba=b, aaaabb=a⟩φ(a) = a, φ(b) = b
1119172⟨a, b | aba=b, aaabba=a⟩φ(a) = a, φ(b) = b
1119182⟨a, b | aba=b, aabbaa=a⟩φ(a) = a, φ(b) = b
1119190⟨a, b | aba=b, abaaab=a⟩φ(a) = a, φ(b) = b