#12897 ⟨a, b | aab=aaa, baaa=b⟩

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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.

1aba2abb2a2bab2b3a2b2ab3a2b3
11aba2abb2a2bab2b3a2b2ab3a2b3
aaa2aba2ba2bab2a2b2a2b2ab3a2b3a2b3a2b
bbb2b2b3b3b3bbbb2b2b3
a2a2a2ba2ba2b2a2b2a2b2a2b3a2b3a2b3a2ba2ba2b2
ababab2ab2ab3ab3ab3abababab2ab2ab3
b2b2b3b3bbbb2b2b2b3b3b
a2ba2ba2b2a2b2a2b3a2b3a2b3a2ba2ba2ba2b2a2b2a2b3
ab2ab2ab3ab3abababab2ab2ab2ab3ab3ab
b3b3bbb2b2b2b3b3b3bbb2
a2b2a2b2a2b3a2b3a2ba2ba2ba2b2a2b2a2b2a2b3a2b3a2b
ab3ab3ababab2ab2ab2ab3ab3ab3ababab2
a2b3a2b3a2ba2ba2b2a2b2a2b2a2b3a2b3a2b3a2ba2ba2b2

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. ba ⇒ b2 [4]
2. a3 ⇒ a2b [1]
3. b4 ⇒ b [5]
# ab:aab=aaa,baaa=b ba - -
ba=bb
aaa=aab
bbbb=b

Same cardinality

37 unique, 654 total

Σ#PresentationPropertiesDescriptionφ
7124⟨a, b | aab=a, bbb=1⟩Noc 1261
8458⟨a, b | aaaa=1, abbb=1⟩Grp Com 12ℤ12325
8645⟨a, b | ab=aa, bbb=b⟩Noc 121
91646⟨a, b | aab=bb, aba=a⟩Noc 122
91963⟨a, b | aba=b, aaabb=1⟩Grp 1230
91998⟨a, b | aaa=a, bbbb=a⟩Noc Com 12ℕ(12 = 4)6
92018⟨a, b | aaa=b, bbbb=a⟩Noc Com 12ℕ(12 = 1)27
92019⟨a, b | aaa=b, bbbb=b⟩Noc Com 12ℕ(12 = 3)23
92259⟨a, b | ab=aa, bbb=bb⟩Noc 12
105040⟨a, b | aaa=aa, bbbb=a⟩Noc Com 12ℕ(12 = 8)
105072⟨a, b | aaa=ab, bbbb=a⟩Noc Com 12ℕ(12 = 5)7
105092⟨a, b | aaa=bb, bbbb=a⟩Noc Com 12ℕ(12 = 2)43
105202⟨a, b | aab=bb, abba=a⟩Noc 1241
105330⟨a, b | aaa=ab, bab=bb⟩Noc 121
106597⟨a, b | aaa=b, bbbb=bb⟩Noc Com 12ℕ(12 = 6)4
106660⟨a, b | aab=a, bbbb=ba⟩Noc 121
106661⟨a, b | aab=a, bbbb=bb⟩Noc 121
107105⟨a, b | ab=aa, bbaa=bb⟩Noc 122
1112910⟨a, b | aab=aaa, bbbb=a⟩Noc Com 12ℕ(12 = 9)2
1115428⟨a, b | aaa=aa, abbbb=b⟩Noc 12
1115996⟨a, b | aaa=ab, aabb=bb⟩Noc 122
1116057⟨a, b | aaa=bb, aabb=ab⟩Noc 124
1116104⟨a, b | aab=aa, abab=bb⟩Noc 12
1116274⟨a, b | aab=bb, aaaa=ab⟩Noc 121
1116275⟨a, b | aab=bb, aaaa=ba⟩Noc 12
1116448⟨a, b | aba=bb, aabb=aa⟩Noc 122
1119773⟨a, b | aba=b, bbbbb=aa⟩Noc Com 12
1119917⟨a, b | aaa=b, bbbb=abb⟩Noc Com 12ℕ(12 = 7)2
1120054⟨a, b | aab=b, aaaa=bba⟩Noc 12
1120686⟨a, b | bb=aa, aaaaab=a⟩Noc Com 1215
1120787⟨a, b | ab=aa, baaaa=bb⟩Noc 127
1121086⟨a, b | bb=aa, aaaa=aba⟩Noc 124
1121092⟨a, b | bb=aa, aaab=aba⟩Noc 123
1121112⟨a, b | bb=aa, abab=aba⟩Noc 12
1124147⟨a, b | aa=a, abbbbbb=b⟩Noc 12
1124991⟨a, b | ab=a, baaaaa=bb⟩Noc 12
1125603⟨a, b | ab=a, bbaaa=bbb⟩Noc 12

Isomorphic instances

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

5 total

Σ#PresentationMapping
1112899⟨a, b | aab=aaa, baab=b⟩φ(a) = a, φ(b) = b
1112901⟨a, b | aab=aaa, baba=b⟩φ(a) = a, φ(b) = b
1112905⟨a, b | aab=aaa, bbaa=b⟩φ(a) = a, φ(b) = b
1112909⟨a, b | aab=aaa, bbba=b⟩φ(a) = a, φ(b) = b
1120023⟨a, b | aab=a, bbaa=bbb⟩φ(a) = bb, φ(b) = a