#16042 ⟨a, b | aaa=ab, bbbb=ab⟩

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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.

1aba2bab2a3ba2b2ab3a4b2a2b3aa5b3a2a6a7a8
11aba2bab2a3ba2b2ab3a4b2a2b3aa5b3a2a6a7a8
aaa2a3a3a4a5a4a5a6a7a5a7a8a6a4a7a8a4
bbbab2ba2b2ab3a5b2a2b3aa3a6b3a2a4a7a5a8a4a5
a2a2a3a4a4a5a6a5a6a7a8a6a8a4a7a5a8a4a5
bababa2a5a5a6a7a6a7a8a4a7a4a5a8a6a4a5a6
b2b2b2ab3b2a2b3aa3a7b3a2a4a5a8a5a6a4a7a5a6a7
a3a3a4a5a5a6a7a6a7a8a4a7a4a5a8a6a4a5a6
ba2ba2a5a6a6a7a8a7a8a4a5a8a5a6a4a7a5a6a7
b2ab2ab2a2a7a7a8a4a8a4a5a6a4a6a7a5a8a6a7a8
b3b3b3aa3b3a2a4a5a4a5a6a7a5a7a8a6a4a7a8a4
a4a4a5a6a6a7a8a7a8a4a5a8a5a6a4a7a5a6a7
b2a2b2a2a7a8a8a4a5a4a5a6a7a5a7a8a6a4a7a8a4
b3ab3ab3a2a4a4a5a6a5a6a7a8a6a8a4a7a5a8a4a5
a5a5a6a7a7a8a4a8a4a5a6a4a6a7a5a8a6a7a8
b3a2b3a2a4a5a5a6a7a6a7a8a4a7a4a5a8a6a4a5a6
a6a6a7a8a8a4a5a4a5a6a7a5a7a8a6a4a7a8a4
a7a7a8a4a4a5a6a5a6a7a8a6a8a4a7a5a8a4a5
a8a8a4a5a5a6a7a6a7a8a4a7a4a5a8a6a4a5a6

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 ⇒ a4 [3]
2. ba3 ⇒ a5 [4]
3. ab ⇒ a3 [1]
4. b4 ⇒ a3 [2]
# ab:aaa=ab,bbbb=ab a/b - -
aaaaaaaaa=aaaa
baaa=aaaaa
ab=aaa
bbbb=aaa

Same cardinality

17 unique, 104 total

Σ#PresentationPropertiesDescriptionφ
81121⟨a, b | aa=1, abbba=b⟩Noc 1817
93387⟨a, b | aa=1, abbbbb=b⟩Noc 1831
105521⟨a, b | aaab=1, bbbbbb=1⟩Grp Com 18ℤ1833
106732⟨a, b | aba=a, aaab=bb⟩Noc 18
106788⟨a, b | aba=b, baab=aa⟩Noc 18
106795⟨a, b | aba=b, bbbb=aa⟩Noc 18
107039⟨a, b | bb=aa, aaaba=b⟩Noc 182
108910⟨a, b | aa=a, bbbbb=ab⟩Noc 181
1112187⟨a, b | aaaa=ab, babb=b⟩Noc 182
1115797⟨a, b | aab=bb, bbbba=a⟩Noc 18
1116043⟨a, b | aaa=ab, bbbb=ba⟩Noc 18
1116313⟨a, b | aab=bb, baba=aa⟩Noc 18
1118758⟨a, b | aaa=a, bbbbbb=a⟩Noc Com 18ℕ(18 = 6)
1118830⟨a, b | aaa=b, bbbbbb=a⟩Noc Com 18ℕ(18 = 1)
1118831⟨a, b | aaa=b, bbbbbb=b⟩Noc Com 18ℕ(18 = 3)
1119624⟨a, b | aab=b, bbbba=aa⟩Noc 18
1120914⟨a, b | bb=aa, ababa=aa⟩Noc 181