#1602 ⟨a, b | aaa=bb, aba=a⟩

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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
  8. Isomorphic instances

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.

1aba2abbab2a2bba2babb3ba2bb4b5b6
11aba2abbab2a2bba2babb3ba2bb4b5b6
aaa2abb2a2bab6b3a2abb2a2bb3b4b5
bbbab2ba2babb6b3ba2bb5b2b4b6b5b6b2
a2a2b2a2bb6b3a2b5b2b2a2bb6b3b2b3b4
ababab6a2abb5b2a2bb4b6b3b5b4b5b6
bababa2babb3ba2bbab2b4ba2babb3ba2bb4b5b6
b2b2b6b3b5b2b2b4b6b6b3b5b2b6b2b3
a2ba2ba2b5b2a2bb4b6b3b3b5b2b4b3b4b5
ba2ba2b3ba2bb2b4ba2b6b3b3ba2bb2b4b3b4b5
babbabbab2ba2babb6b3ba2bb5b2b4b6b5b6b2
b3b3b2b4b6b3b3b5b2b2b4b6b3b2b3b4
ba2bba2bba2b6b3ba2bb5b2b4b4b6b3b5b4b5b6
b4b4b3b5b2b4b4b6b3b3b5b2b4b3b4b5
b5b5b4b6b3b5b5b2b4b4b6b3b5b4b5b6
b6b6b5b2b4b6b6b3b5b5b2b4b6b5b6b2

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 ⇒ b2 [8]
2. ab2 ⇒ b6 [7]
3. b2a ⇒ b6 [9]
4. aba ⇒ a [2]
5. a3 ⇒ b2 [1]
# ab:aaa=bb,aba=a b/a - -
bbbbbbb=bb
abb=bbbbbb
bba=bbbbbb
aba=a
aaa=bb

Same cardinality

22 unique, 188 total

Σ#PresentationPropertiesDescriptionφ
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
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

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

2 total

Σ#PresentationMapping
1119642⟨a, b | aba=a, aaaba=bb⟩φ(a) = a, φ(b) = b
1119649⟨a, b | aba=a, aabaa=bb⟩φ(a) = a, φ(b) = b