#13123 ⟨a, b | aab=aaa, aba=bb⟩

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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.

1aba2abbaa3ababa2baba4aba2(ab)2ba3aba3
11aba2abbaa3ababa2baba4aba2(ab)2ba3aba3
aaa2aba3a3abaa4a4aba2(ab)2a4a4a4aba3a4
bbbaababa2bababa2ba3(ab)2aba3aba3a4a4a4a4a4
a2a2a3a3a4a4a4a4a4a4a4a4a4a4a4a4
abababaa4aba2(ab)2a4aba3a4a4a4a4a4a4a4a4
bababa2babba3ba3(ab)2a4a4a4a4a4a4a4a4a4
a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4a4
abaabaaba2(ab)2aba3aba3a4a4a4a4a4a4a4a4a4a4
ba2ba2ba3ba3a4a4a4a4a4a4a4a4a4a4a4a4
babbab(ab)2a4a4a4a4a4a4a4a4a4a4a4a4a4
a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
aba2aba2aba3aba3a4a4a4a4a4a4a4a4a4a4a4a4
(ab)2(ab)2a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba3ba3a4a4a4a4a4a4a4a4a4a4a4a4a4a4
aba3aba3a4a4a4a4a4a4a4a4a4a4a4a4a4a4

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. a5 ⇒ a4 [4]
2. a2b ⇒ a3 [1]
3. ba4 ⇒ a4 [6]
4. b2 ⇒ aba [2]
5. (ba)2 ⇒ (ab)2 [3]
# ab:aab=aaa,aba=bb reversed:a/b - -
aaaaa=aaaa
aab=aaa
baaaa=aaaa
bb=aba
baba=abab

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
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
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