#5111 ⟨a, b | aab=aa, baaa=b⟩

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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.

1aba2abbaa3ababa2a4aba2
11aba2abbaa3ababa2a4aba2
aaa2aba3a2abaa4a3aba2a2a4
bbbabba2bababba2ba2bab
a2a2a3a2a4a3a3a2a4a4a3a2
abababaababa2abaabaababa2aba2abaab
bababa2babba2ba2babbba2ba
a3a3a4a3a2a4a4a3a2a2a4a3
abaabaaba2abaababa2aba2abaabababa2aba
ba2ba2bba2babbba2bababba2
a4a4a2a4a3a2a2a4a3a3a2a4
aba2aba2ababa2abaabababa2abaabaababa2

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. b2 ⇒ b [4]
2. a2b ⇒ a2 [1]
3. bab ⇒ ba [5]
4. ba3 ⇒ b [2]
5. a5 ⇒ a2 [3]
# ab:aab=aa,baaa=b ab - -
bb=b
aab=aa
bab=ba
baaa=b
aaaaa=aa

Same cardinality

29 unique, 473 total

Σ#PresentationPropertiesDescriptionφ
91472⟨a, b | aaa=bb, abbb=1⟩Grp Com 11ℤ11259
104155⟨a, b | abb=aaa, bba=b⟩Noc 114
105086⟨a, b | aaa=bb, abbb=a⟩Noc Com 1111
105087⟨a, b | aaa=bb, abbb=b⟩Noc Com 113
105322⟨a, b | aaa=ab, abb=bb⟩Noc 113
105335⟨a, b | aaa=ab, bbb=aa⟩Noc 113
105410⟨a, b | aab=bb, bab=aa⟩Noc 112
106292⟨a, b | aaa=b, aabbb=a⟩Noc Com 11ℕ(11 = 1)35
106293⟨a, b | aaa=b, aabbb=b⟩Noc Com 11ℕ(11 = 3)11
106380⟨a, b | aab=a, bbbbb=a⟩Noc Com 11ℕ(11 = 5)1
106834⟨a, b | aba=b, abb=aaa⟩Noc 11
107114⟨a, b | ab=aa, bbbb=aa⟩Noc 111
107116⟨a, b | ab=aa, bbbb=ba⟩Noc 11
108691⟨a, b | aa=b, abbbbb=b⟩Noc Com 11ℕ(11 = 2)32
108777⟨a, b | ab=a, baaaaa=b⟩Noc 1125
109083⟨a, b | ab=a, bbaaa=bb⟩Noc 114
1112179⟨a, b | aaaa=ab, abbb=b⟩Noc Com 11ℕ(11 = 4)20
1115607⟨a, b | aab=aa, bbbbb=a⟩Noc Com 11ℕ(11 = 10)1
1115671⟨a, b | aab=ab, bbbbb=a⟩Noc Com 11ℕ(11 = 6)8
1116148⟨a, b | aab=ab, aaaa=bb⟩Noc 111
1116212⟨a, b | aab=ba, aaaa=bb⟩Noc 11
1118715⟨a, b | aaa=a, aabbba=b⟩Noc 11
1119070⟨a, b | aab=b, bbabbb=a⟩Noc Com 114
1120096⟨a, b | aab=b, abba=aaa⟩Noc 111
1120112⟨a, b | aab=b, baaa=aaa⟩Noc 11
1120168⟨a, b | aab=b, bbbb=aaa⟩Noc Com 11
1120885⟨a, b | bb=aa, aaaaa=ab⟩Noc 1115
1125539⟨a, b | ab=a, baaaa=bbb⟩Noc 11
1125635⟨a, b | ab=a, bbbaa=bbb⟩Noc 11

Isomorphic instances

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

7 total

Σ#PresentationMapping
1112227⟨a, b | aaab=aa, baaa=b⟩φ(a) = a, φ(b) = baa
1112337⟨a, b | aaba=aa, baaa=b⟩φ(a) = a, φ(b) = baa
1115580⟨a, b | aab=aa, baaab=b⟩φ(a) = a, φ(b) = b
1115582⟨a, b | aab=aa, baaba=b⟩φ(a) = a, φ(b) = b
1115586⟨a, b | aab=aa, babaa=b⟩φ(a) = a, φ(b) = b
1115594⟨a, b | aab=aa, bbaaa=b⟩φ(a) = a, φ(b) = b
1119475⟨a, b | aab=a, bbaaa=bb⟩φ(a) = baa, φ(b) = a