#16142 ⟨a, b | aab=aa, 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.
The zero element z satisfies zy = yz = z 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.

1abbab2b2ab3b3ab4b4ab5b5ab6b6ab7
11abbab2b2ab3b3ab4b4ab5b5ab6b6ab7
aab7b4b4ab5b5ab6b6ab7b7b7b7b7b7b7
bbbab2b2ab3b3ab4b4ab5b5ab6b6ab7b7b7
babab7b5b5ab6b6ab7b7b7b7b7b7b7b7b7
b2b2b2ab3b3ab4b4ab5b5ab6b6ab7b7b7b7b7
b2ab2ab7b6b6ab7b7b7b7b7b7b7b7b7b7b7
b3b3b3ab4b4ab5b5ab6b6ab7b7b7b7b7b7b7
b3ab3ab7b7b7b7b7b7b7b7b7b7b7b7b7b7
b4b4b4ab5b5ab6b6ab7b7b7b7b7b7b7b7b7
b4ab4ab7b7b7b7b7b7b7b7b7b7b7b7b7b7
b5b5b5ab6b6ab7b7b7b7b7b7b7b7b7b7b7
b5ab5ab7b7b7b7b7b7b7b7b7b7b7b7b7b7
b6b6b6ab7b7b7b7b7b7b7b7b7b7b7b7b7
b6ab6ab7b7b7b7b7b7b7b7b7b7b7b7b7b7
b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7b7

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. b8 ⇒ b7 [4]
2. ab ⇒ b4 [2]
3. b7a ⇒ b7 [5]
4. a2 ⇒ b7 [3]
# ab:aab=aa,bbbb=ab b/a - -
bbbbbbbb=bbbbbbb
ab=bbbb
bbbbbbba=bbbbbbb
aa=bbbbbbb

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
1116041⟨a, b | aaa=ab, bbbb=aa⟩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