#16041 ⟨a, b | aaa=ab, bbbb=aa⟩

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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

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.

1aba2bab2a3b2ab3a4b3aa5a6a7a8
11aba2bab2a3b2ab3a4b3aa5a6a7a8
aaa2a3a3a4a5a4a6a7a5a8a6a7a8a3
bbbab2a4b2ab3a5b3aa2a6a3a7a8a3a4
a2a2a3a4a4a5a6a5a7a8a6a3a7a8a3a4
babaa4a5a5a6a7a6a8a3a7a4a8a3a4a5
b2b2b2ab3a6b3aa2a7a3a4a8a5a3a4a5a6
a3a3a4a5a5a6a7a6a8a3a7a4a8a3a4a5
b2ab2aa6a7a7a8a3a8a4a5a3a6a4a5a6a7
b3b3b3aa2a8a3a4a3a5a6a4a7a5a6a7a8
a4a4a5a6a6a7a8a7a3a4a8a5a3a4a5a6
b3ab3aa8a3a3a4a5a4a6a7a5a8a6a7a8a3
a5a5a6a7a7a8a3a8a4a5a3a6a4a5a6a7
a6a6a7a8a8a3a4a3a5a6a4a7a5a6a7a8
a7a7a8a3a3a4a5a4a6a7a5a8a6a7a8a3
a8a8a3a4a4a5a6a5a7a8a6a3a7a8a3a4

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. a9 ⇒ a3 [3]
2. ba2 ⇒ a4 [4]
3. ab ⇒ a3 [1]
4. b4 ⇒ a2 [2]
# ab:aaa=ab,bbbb=aa a/b - -
aaaaaaaaa=aaa
baa=aaaa
ab=aaa
bbbb=aa

Same cardinality

22 unique, 190 total

Σ#PresentationPropertiesDescriptionφ
91602⟨a, b | aaa=bb, aba=a⟩Noc 152
91603⟨a, b | aaa=bb, aba=b⟩Noc 151
91751⟨a, b | aaab=1, bbbbb=1⟩Grp Com 15ℤ15138
91993⟨a, b | aaa=a, abbb=b⟩Noc 154
105342⟨a, b | aaa=bb, aba=aa⟩Noc 151
105344⟨a, b | aaa=bb, aba=bb⟩Noc 152
106276⟨a, b | aaa=a, bbbbb=a⟩Noc Com 15ℕ(15 = 5)
106316⟨a, b | aaa=b, bbbbb=a⟩Noc Com 15ℕ(15 = 1)
106317⟨a, b | aaa=b, bbbbb=b⟩Noc Com 15ℕ(15 = 3)
1112981⟨a, b | abb=aaa, baaa=b⟩Noc 153
1112985⟨a, b | abb=aaa, baba=b⟩Noc 157
1113123⟨a, b | aab=aaa, aba=bb⟩Noc 15
1113191⟨a, b | abb=aaa, bbb=ab⟩Noc 151
1113192⟨a, b | abb=aaa, bbb=ba⟩Noc 15
1115439⟨a, b | aaa=aa, bbbbb=a⟩Noc Com 15ℕ(15 = 10)
1115503⟨a, b | aaa=ab, bbbbb=a⟩Noc Com 15ℕ(15 = 6)1
1115543⟨a, b | aaa=bb, bbbbb=a⟩Noc Com 15ℕ(15 = 2)1
1116142⟨a, b | aab=aa, bbbb=ab⟩Noc 15
1119374⟨a, b | aaa=b, bbbbb=ab⟩Noc Com 15ℕ(15 = 4)
1119502⟨a, b | aab=a, bbbbb=ba⟩Noc 151
1120144⟨a, b | aab=b, bbaa=aaa⟩Noc 15
1120819⟨a, b | ab=aa, bbaaa=bb⟩Noc 156