#20991 ⟨a, b | ab=aa, baaa=bbb⟩

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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.
A left zero element x satisfies xy = x for all y.
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.

1aba2bab2a3ba2b2aa4ba3b2a2ba4
11aba2bab2a3ba2b2aa4ba3b2a2ba4
aaa2a2a3a3a3a4a4a4a4a4a4a4
bbbab2ba2b2aba3ba3b2a2ba4ba4ba4ba4ba4
a2a2a3a3a4a4a4a4a4a4a4a4a4a4
bababa2ba2ba3ba3ba3ba4ba4ba4ba4ba4ba4ba4
b2b2b2aba3b2a2ba4ba4ba4ba4ba4ba4ba4ba4ba4
a3a3a4a4a4a4a4a4a4a4a4a4a4a4
ba2ba2ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2ab2ab2a2b2a2ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2a2b2a2ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4

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. ab ⇒ a2 [1]
3. b2a3 ⇒ ba4 [4]
4. b3 ⇒ ba3 [2]
# ab:ab=aa,baaa=bbb a/b - -
aaaaa=aaaa
ab=aa
bbaaa=baaaa
bbb=baaa

Same cardinality

26 unique, 300 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
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

Isomorphic instances

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

3 total

Σ#PresentationMapping
1120999⟨a, b | ab=aa, baab=bbb⟩φ(a) = a, φ(b) = b
1121007⟨a, b | ab=aa, baba=bbb⟩φ(a) = a, φ(b) = b
1121015⟨a, b | ab=aa, babb=bbb⟩φ(a) = a, φ(b) = b