#11127 ⟨a, b | aaaaa=b, abbbb=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

1aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20
11aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a20
aaa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a201
a2a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a201a
a3a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a201aa2
a4a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a201aa2a3
a5a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19a201aa2a3a4
a6a6a7a8a9a10a11a12a13a14a15a16a17a18a19a201aa2a3a4a5
a7a7a8a9a10a11a12a13a14a15a16a17a18a19a201aa2a3a4a5a6
a8a8a9a10a11a12a13a14a15a16a17a18a19a201aa2a3a4a5a6a7
a9a9a10a11a12a13a14a15a16a17a18a19a201aa2a3a4a5a6a7a8
a10a10a11a12a13a14a15a16a17a18a19a201aa2a3a4a5a6a7a8a9
a11a11a12a13a14a15a16a17a18a19a201aa2a3a4a5a6a7a8a9a10
a12a12a13a14a15a16a17a18a19a201aa2a3a4a5a6a7a8a9a10a11
a13a13a14a15a16a17a18a19a201aa2a3a4a5a6a7a8a9a10a11a12
a14a14a15a16a17a18a19a201aa2a3a4a5a6a7a8a9a10a11a12a13
a15a15a16a17a18a19a201aa2a3a4a5a6a7a8a9a10a11a12a13a14
a16a16a17a18a19a201aa2a3a4a5a6a7a8a9a10a11a12a13a14a15
a17a17a18a19a201aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16
a18a18a19a201aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17
a19a19a201aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18
a20a201aa2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18a19

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. a21 ⇒ 1 [2]
2. b ⇒ a5 [1]
# ab:aaaaa=b,abbbb=1 a/b - -
aaaaaaaaaaaaaaaaaaaaa=1
b=aaaaa

Same cardinality

8 unique, 51 total

Σ#PresentationPropertiesφ
8402⟨a, b | aaa=ab, bbb=1⟩Noc 2139
91599⟨a, b | aaa=ab, bbb=b⟩Noc 211
105309⟨a, b | aaa=aa, bbb=ab⟩Noc 21
106265⟨a, b | aaa=a, abbbb=b⟩Noc 212
1113091⟨a, b | bab=aaa, abba=b⟩Noc 21
1113248⟨a, b | bab=aaa, bbb=aa⟩Noc 21
1115158⟨a, b | aab=ba, ababab=1⟩Grp 211
1119212⟨a, b | aba=b, baaaab=a⟩Noc 21

Isomorphic instances

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

13 total

Σ#PresentationMapping
1111131⟨a, b | aaaaa=b, babbb=1⟩φ(a) = bbbbbbbbbbbbbbbbb, φ(b) = b
1111132⟨a, b | aaaaa=b, bbabb=1⟩φ(a) = bbbbbbbbbbbbbbbbb, φ(b) = b
1113862⟨a, b | aaaa=b, abbbbb=1⟩φ(a) = bbbbbbbbbbbbbbbb, φ(b) = b
1113869⟨a, b | aaaa=b, babbbb=1⟩φ(a) = bbbbbbbbbbbbbbbb, φ(b) = b
1113871⟨a, b | aaaa=b, bbabbb=1⟩φ(a) = bbbbbbbbbbbbbbbb, φ(b) = b
1116759⟨a, b | aaab=1, bbbbbbb=1⟩φ(a) = b, φ(b) = bbbbbbbbbbbbbbbbbb
1116887⟨a, b | aaba=1, bbbbbbb=1⟩φ(a) = b, φ(b) = bbbbbbbbbbbbbbbbbb
1118287⟨a, b | aaa=b, bbbbbbb=1⟩φ(a) = b, φ(b) = bbb
1121219⟨a, b | aaa=1, abbbbbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1121245⟨a, b | aaa=1, babbbbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1121252⟨a, b | aaa=1, bbabbbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1121254⟨a, b | aaa=1, bbbabbbb=1⟩φ(a) = bbbbbbbbbbbbbb, φ(b) = b
1121790⟨a, b | aaa=1, bbbbbbb=a⟩φ(a) = bbbbbbb, φ(b) = b