#13372 ⟨a, b | aaaab=1, bbbbbb=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

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a23
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a23
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231a
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231aa2
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231aa2a3
a5a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231aa2a3a4
a6a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231aa2a3a4a5
a7a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231aa2a3a4a5a6
a8a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231aa2a3a4a5a6a7
a9a9a10a11a12a13a14a15a16a17a18a19a20a21a22a231aa2a3a4a5a6a7a8
a10a10a11a12a13a14a15a16a17a18a19a20a21a22a231aa2a3a4a5a6a7a8a9
a11a11a12a13a14a15a16a17a18a19a20a21a22a231aa2a3a4a5a6a7a8a9a10
a12a12a13a14a15a16a17a18a19a20a21a22a231aa2a3a4a5a6a7a8a9a10a11
a13a13a14a15a16a17a18a19a20a21a22a231aa2a3a4a5a6a7a8a9a10a11a12
a14a14a15a16a17a18a19a20a21a22a231aa2a3a4a5a6a7a8a9a10a11a12a13
a15a15a16a17a18a19a20a21a22a231aa2a3a4a5a6a7a8a9a10a11a12a13a14
a16a16a17a18a19a20a21a22a231aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
a17a17a18a19a20a21a22a231aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
a18a18a19a20a21a22a231aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
a19a19a20a21a22a231aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18
a20a20a21a22a231aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19
a21a21a22a231aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20
a22a22a231aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21
a23a231aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20a21a22

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. a24 ⇒ 1 [9]
2. b ⇒ a20 [8]
# ab:aaaab=1,bbbbbb=1 a/b - -
aaaaaaaaaaaaaaaaaaaaaaaa=1
b=aaaaaaaaaaaaaaaaaaaa

Same cardinality

6 unique, 69 total

Σ#PresentationPropertiesφ
8425⟨a, b | aab=ba, bbb=1⟩Noc 2421
105009⟨a, b | aba=bb, aabbb=1⟩Grp 2430
106935⟨a, b | bb=aa, aaabab=1⟩Grp 242
1111515⟨a, b | ababa=b, abbaa=1⟩Grp 2410
1119638⟨a, b | aba=a, aaaab=bb⟩Noc 24
1120252⟨a, b | aba=b, aaaa=bab⟩Noc 24

Isomorphic instances

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

8 total

Σ#PresentationMapping
1113436⟨a, b | aaaba=1, bbbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaaaaaaaaaa
1113536⟨a, b | aabaa=1, bbbbbb=1⟩φ(a) = a, φ(b) = aaaaaaaaaaaaaaaaaaaa
1113872⟨a, b | aaaa=b, bbbbbb=1⟩φ(a) = a, φ(b) = aaaa
1116611⟨a, b | aaaa=1, abbbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaaaa, φ(b) = a
1116625⟨a, b | aaaa=1, babbbbb=1⟩φ(a) = aaaaaaaaaaaaaaaaaa, φ(b) = a
1116629⟨a, b | aaaa=1, bbabbbb=1⟩φ(a) = aaaaaaaaaaaaaaaaaa, φ(b) = a
1116630⟨a, b | aaaa=1, bbbabbb=1⟩φ(a) = aaaaaaaaaaaaaaaaaa, φ(b) = a
1117158⟨a, b | aaaa=1, bbbbbb=a⟩φ(a) = aaaaaa, φ(b) = a