#20723 ⟨a, b | ab=aa, aaaaa=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
  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.

1aba2baa3ba2a4a5
11aba2baa3ba2a4a5
aaa2a2a3a3a4a4a5a3
bbbaa5ba2a3a4a4a5a3
a2a2a3a3a4a4a5a5a3a4
bababa2ba2a4a4a5a5a3a4
a3a3a4a4a5a5a3a3a4a5
ba2ba2a4a4a5a5a3a3a4a5
a4a4a5a5a3a3a4a4a5a3
a5a5a3a3a4a4a5a5a3a4

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. a6 ⇒ a3 [3]
2. ba3 ⇒ a4 [5]
3. ab ⇒ a2 [1]
4. b2 ⇒ a5 [2]
# ab:ab=aa,aaaaa=bb a/b - -
aaaaaa=aaa
baaa=aaaa
ab=aa
bb=aaaaa

Same cardinality

26 unique, 927 total

Σ#PresentationPropertiesDescriptionφ
7118⟨a, b | aaa=b, bbb=1⟩Grp Com 9ℤ9562
8568⟨a, b | aaa=a, abb=b⟩Noc 95
8571⟨a, b | aaa=a, bbb=a⟩Noc Com 9ℕ(9 = 3)48
8581⟨a, b | aaa=b, bbb=a⟩Noc Com 9ℕ(9 = 1)62
8601⟨a, b | aba=b, bab=a⟩Noc 93
91584⟨a, b | aaa=aa, bbb=a⟩Noc Com 9ℕ(9 = 6)16
91595⟨a, b | aaa=ab, bab=b⟩Noc 99
91598⟨a, b | aaa=ab, bbb=a⟩Noc Com 9ℕ(9 = 4)40
91608⟨a, b | aaa=bb, bbb=a⟩Noc Com 9ℕ(9 = 2)70
91991⟨a, b | aaa=a, abba=b⟩Noc 93
92064⟨a, b | aab=b, abba=a⟩Noc 938
92255⟨a, b | ab=aa, bba=bb⟩Noc 9
104125⟨a, b | aab=aaa, baa=b⟩Noc 92
104130⟨a, b | aab=aaa, bbb=a⟩Noc Com 9ℕ(9 = 7)1
104156⟨a, b | abb=aaa, bbb=a⟩Noc Com 9ℕ(9 = 5)12
105035⟨a, b | aaa=aa, abbb=b⟩Noc 93
105079⟨a, b | aaa=bb, aaba=b⟩Noc Com 98
105124⟨a, b | aab=aa, bbbb=a⟩Noc Com 9ℕ(9 = 8)5
105339⟨a, b | aaa=bb, aab=ab⟩Noc 91
106824⟨a, b | aab=b, bbb=aaa⟩Noc Com 95
107139⟨a, b | bb=aa, aaaa=ab⟩Noc 93
109327⟨a, b | ab=a, baaa=bbb⟩Noc 93
109359⟨a, b | ab=a, bbaa=bbb⟩Noc 92
1119506⟨a, b | aab=b, aaaaa=ba⟩Noc 9
1119507⟨a, b | aab=b, aaaaa=bb⟩Noc Com 9
1125656⟨a, b | ab=a, bbbbb=baa⟩Noc 9

Isomorphic instances

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

15 total

Σ#PresentationMapping
1120727⟨a, b | ab=aa, aaaab=bb⟩φ(a) = a, φ(b) = b
1120731⟨a, b | ab=aa, aaaba=bb⟩φ(a) = a, φ(b) = b
1120735⟨a, b | ab=aa, aaabb=bb⟩φ(a) = a, φ(b) = b
1120739⟨a, b | ab=aa, aabaa=bb⟩φ(a) = a, φ(b) = b
1120743⟨a, b | ab=aa, aabab=bb⟩φ(a) = a, φ(b) = b
1120747⟨a, b | ab=aa, aabba=bb⟩φ(a) = a, φ(b) = b
1120751⟨a, b | ab=aa, aabbb=bb⟩φ(a) = a, φ(b) = b
1120755⟨a, b | ab=aa, abaaa=bb⟩φ(a) = a, φ(b) = b
1120759⟨a, b | ab=aa, abaab=bb⟩φ(a) = a, φ(b) = b
1120763⟨a, b | ab=aa, ababa=bb⟩φ(a) = a, φ(b) = b
1120767⟨a, b | ab=aa, ababb=bb⟩φ(a) = a, φ(b) = b
1120771⟨a, b | ab=aa, abbaa=bb⟩φ(a) = a, φ(b) = b
1120775⟨a, b | ab=aa, abbab=bb⟩φ(a) = a, φ(b) = b
1120779⟨a, b | ab=aa, abbba=bb⟩φ(a) = a, φ(b) = b
1120783⟨a, b | ab=aa, abbbb=bb⟩φ(a) = a, φ(b) = b