#13192 ⟨a, b | abb=aaa, bbb=ba⟩

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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.

1aba2abb2a2bab2b3ab3b4ab4b5ab5b6
11aba2abb2a2bab2b3ab3b4ab4b5ab5b6
aaa2abab2a2bab2ab3ab4ab3ab5ab4ab4ab5ab5ab4
bbb3b2b5b4b3b6b5b4b6b5b5b6b6b5
a2a2ab2a2bab4ab3ab4ab5ab4ab5ab5ab4ab4ab5ab5ab4
ababab3ab2ab5ab4ab3ab4ab5ab4ab4ab5ab5ab4ab4ab5
b2b2b4b3b6b5b4b5b6b5b5b6b6b5b5b6
a2ba2bab5ab4ab5ab4ab5ab4ab5ab4ab4ab5ab5ab4ab4ab5
ab2ab2ab4ab3ab4ab5ab4ab5ab4ab5ab5ab4ab4ab5ab5ab4
b3b3b5b4b5b6b5b6b5b6b6b5b5b6b6b5
ab3ab3ab5ab4ab5ab4ab5ab4ab5ab4ab4ab5ab5ab4ab4ab5
b4b4b6b5b6b5b6b5b6b5b5b6b6b5b5b6
ab4ab4ab4ab5ab4ab5ab4ab5ab4ab5ab5ab4ab4ab5ab5ab4
b5b5b5b6b5b6b5b6b5b6b6b5b5b6b6b5
ab5ab5ab5ab4ab5ab4ab5ab4ab5ab4ab4ab5ab5ab4ab4ab5
b6b6b6b5b6b5b6b5b6b5b5b6b6b5b5b6

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. b7 ⇒ b5 [4]
2. ab6 ⇒ ab4 [5]
3. ba ⇒ b3 [2]
4. a2b2 ⇒ ab4 [3]
5. a3 ⇒ ab2 [1]
# ab:abb=aaa,bbb=ba b/a - -
bbbbbbb=bbbbb
abbbbbb=abbbb
ba=bbb
aabb=abbbb
aaa=abb

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
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
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