#13340 ⟨a, b | aaaab=1, abbbbb=1⟩

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Contents

  1. Properties
  2. Commutative structure
  3. Finite structure
  4. Staircase diagram
  5. Cayley table
  6. Right Cayley graph
  7. Rewriting system
  8. Same cardinality
  9. Isomorphic instances

Properties

Commutative structure

Finite structure

The order of x is the least n such that xn = 1.

Staircase diagram

Cayley table

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a181
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a181a
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a181aa2
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a17a181aa2a3
a5a5a6a7a8a9a10a11a12a13a14a15a16a17a181aa2a3a4
a6a6a7a8a9a10a11a12a13a14a15a16a17a181aa2a3a4a5
a7a7a8a9a10a11a12a13a14a15a16a17a181aa2a3a4a5a6
a8a8a9a10a11a12a13a14a15a16a17a181aa2a3a4a5a6a7
a9a9a10a11a12a13a14a15a16a17a181aa2a3a4a5a6a7a8
a10a10a11a12a13a14a15a16a17a181aa2a3a4a5a6a7a8a9
a11a11a12a13a14a15a16a17a181aa2a3a4a5a6a7a8a9a10
a12a12a13a14a15a16a17a181aa2a3a4a5a6a7a8a9a10a11
a13a13a14a15a16a17a181aa2a3a4a5a6a7a8a9a10a11a12
a14a14a15a16a17a181aa2a3a4a5a6a7a8a9a10a11a12a13
a15a15a16a17a181aa2a3a4a5a6a7a8a9a10a11a12a13a14
a16a16a17a181aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
a17a17a181aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
a18a181aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17

Right Cayley graph

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. a19 ⇒ 1 [8]
2. b ⇒ a15 [7]
# ab:aaaab=1,abbbbb=1 a/b - -
aaaaaaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaaaaa

Same cardinality

5 unique, 9 total

Σ#PresentationPropertiesφ
106263⟨a, b | aaa=a, abbba=b⟩Noc 19
106838⟨a, b | aba=b, bab=aaa⟩Noc 19
1114501⟨a, b | aaab=b, abbba=a⟩Noc 193
1116036⟨a, b | aaa=ab, bbab=bb⟩Noc 191
1116053⟨a, b | aaa=bb, aaba=ab⟩Noc 19

Isomorphic instances

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

20 total

Σ#PresentationMapping
1113356⟨a, b | aaaab=1, babbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113364⟨a, b | aaaab=1, bbabbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113368⟨a, b | aaaab=1, bbbabb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113370⟨a, b | aaaab=1, bbbbab=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113371⟨a, b | aaaab=1, bbbbba=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113404⟨a, b | aaaba=1, abbbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113420⟨a, b | aaaba=1, babbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113428⟨a, b | aaaba=1, bbabbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113432⟨a, b | aaaba=1, bbbabb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113434⟨a, b | aaaba=1, bbbbab=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113435⟨a, b | aaaba=1, bbbbba=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113526⟨a, b | aabaa=1, abbbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113533⟨a, b | aabaa=1, babbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1113535⟨a, b | aabaa=1, bbabbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1117286⟨a, b | aaab=1, bbbbbb=a⟩φ(a) = bbbbbb, φ(b) = b
1117414⟨a, b | aaba=1, bbbbbb=a⟩φ(a) = bbbbbb, φ(b) = b
1118267⟨a, b | aaa=b, abbbbbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
1118281⟨a, b | aaa=b, babbbbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
1118285⟨a, b | aaa=b, bbabbbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b
1118286⟨a, b | aaa=b, bbbabbb=1⟩φ(a) = bbbbbbbbbbbbb, φ(b) = b