#124 ⟨a, b | aab=a, bbb=1⟩

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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
  9. Anti-isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
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.

1aba2bab2a3ba2b2aba3b2a2b2a3
11aba2bab2a3ba2b2aba3b2a2b2a3
aaa2a3a3aa2aa2a3a3aa2
bbbab2ba2b2a1ba3b2a2ab2a3a2a3
a2a2a3aaa2a3a2a3aaa2a3
bababa2ba3ba3baba2baba2ba3ba3baba2
b2b2b2a1b2a2abb2a3a2baa3ba2ba3
a3a3aa2a2a3aa3aa2a2a3a
ba2ba2ba3bababa2ba3ba2ba3bababa2ba3
b2ab2ab2a2b2a3b2a3b2ab2a2b2ab2a2b2a3b2a3b2ab2a2
ba3ba3baba2ba2ba3baba3baba2ba2ba3ba
b2a2b2a2b2a3b2ab2ab2a2b2a3b2a2b2a3b2ab2ab2a2b2a3
b2a3b2a3b2ab2a2b2a2b2a3b2ab2a3b2ab2a2b2a2b2a3b2a

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. a4 ⇒ a [5]
2. ab ⇒ a3 [4]
3. b3 ⇒ 1 [2]
# ab:aab=a,bbb=1 a/b - -
aaaa=a
ab=aaa
bbb=1

Same cardinality

37 unique, 598 total

Σ#PresentationPropertiesDescriptionφ
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
1112897⟨a, b | aab=aaa, baaa=b⟩Noc 125
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.

23 total

Σ#PresentationMapping
7150⟨a, b | ab=aa, bbb=1⟩φ(a) = aa, φ(b) = b
8440⟨a, b | aba=ab, bbb=1⟩φ(a) = ba, φ(b) = b
8446⟨a, b | abb=aa, bbb=1⟩φ(a) = a, φ(b) = b
91295⟨a, b | abb=aab, bbb=1⟩φ(a) = aa, φ(b) = b
91299⟨a, b | abb=aba, bbb=1⟩φ(a) = aa, φ(b) = b
92580⟨a, b | aaa=1, abba=ab⟩φ(a) = b, φ(b) = a
107493⟨a, b | aaa=1, aaabba=b⟩φ(a) = b, φ(b) = a
107762⟨a, b | aaa=1, aaaba=bb⟩φ(a) = b, φ(b) = aa
107765⟨a, b | aaa=1, aaabb=ba⟩φ(a) = b, φ(b) = aa
107787⟨a, b | aaa=1, ababa=ab⟩φ(a) = b, φ(b) = aa
108036⟨a, b | aaa=1, aaab=bba⟩φ(a) = b, φ(b) = a
108056⟨a, b | aaa=1, abab=aba⟩φ(a) = b, φ(b) = ba
1121669⟨a, b | aaa=1, aaababa=b⟩φ(a) = b, φ(b) = aa
1121673⟨a, b | aaa=1, aaabbaa=b⟩φ(a) = b, φ(b) = aa
1122210⟨a, b | aaa=1, aaabaa=bb⟩φ(a) = b, φ(b) = a
1122213⟨a, b | aaa=1, aaabab=ba⟩φ(a) = b, φ(b) = ba
1122255⟨a, b | aaa=1, abaaba=ab⟩φ(a) = b, φ(b) = ba
1122739⟨a, b | aaa=1, aaaba=bab⟩φ(a) = b, φ(b) = ba
1122746⟨a, b | aaa=1, aaabb=baa⟩φ(a) = b, φ(b) = a
1122765⟨a, b | aaa=1, aabba=aab⟩φ(a) = b, φ(b) = a
1122782⟨a, b | aaa=1, abaab=aba⟩φ(a) = b, φ(b) = a
1123269⟨a, b | aaa=1, aabb=aaba⟩φ(a) = b, φ(b) = aa
1123277⟨a, b | aaa=1, abab=abaa⟩φ(a) = b, φ(b) = aa

Anti-isomorphic instances

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

38 total

Σ#PresentationMapping
8743⟨a, b | aaa=1, aabb=b⟩φ(a) = bb, φ(b) = a
8745⟨a, b | aaa=1, abab=b⟩φ(a) = bb, φ(b) = a
91308⟨a, b | bab=aab, bbb=1⟩φ(a) = a, φ(b) = bb
92427⟨a, b | aaa=1, aabab=b⟩φ(a) = b, φ(b) = ba
92433⟨a, b | aaa=1, abaab=b⟩φ(a) = b, φ(b) = ba
92572⟨a, b | aaa=1, aabb=ab⟩φ(a) = b, φ(b) = a
92587⟨a, b | aaa=1, baab=ab⟩φ(a) = b, φ(b) = a
107487⟨a, b | aaa=1, aaaabb=b⟩φ(a) = b, φ(b) = a
107497⟨a, b | aaa=1, aabaab=b⟩φ(a) = b, φ(b) = a
107511⟨a, b | aaa=1, abaaab=b⟩φ(a) = b, φ(b) = a
107758⟨a, b | aaa=1, aaaab=bb⟩φ(a) = bb, φ(b) = a
107764⟨a, b | aaa=1, aaabb=ab⟩φ(a) = bb, φ(b) = a
107771⟨a, b | aaa=1, aabab=ab⟩φ(a) = bb, φ(b) = a
107776⟨a, b | aaa=1, aabba=ba⟩φ(a) = bb, φ(b) = a
107805⟨a, b | aaa=1, baaab=ab⟩φ(a) = bb, φ(b) = a
108033⟨a, b | aaa=1, aaab=abb⟩φ(a) = b, φ(b) = a
108044⟨a, b | aaa=1, aaba=bba⟩φ(a) = b, φ(b) = a
108055⟨a, b | aaa=1, abab=aab⟩φ(a) = b, φ(b) = ba
108077⟨a, b | aaa=1, baab=aab⟩φ(a) = b, φ(b) = ba
1121655⟨a, b | aaa=1, aaaaabb=b⟩φ(a) = bb, φ(b) = a
1121659⟨a, b | aaa=1, aaaabab=b⟩φ(a) = bb, φ(b) = a
1121681⟨a, b | aaa=1, aabaaab=b⟩φ(a) = bb, φ(b) = a
1121709⟨a, b | aaa=1, abaaaab=b⟩φ(a) = bb, φ(b) = a
1122198⟨a, b | aaa=1, aaaaab=bb⟩φ(a) = b, φ(b) = a
1122212⟨a, b | aaa=1, aaabab=ab⟩φ(a) = b, φ(b) = ba
1122224⟨a, b | aaa=1, aabaab=ab⟩φ(a) = b, φ(b) = ba
1122229⟨a, b | aaa=1, aababa=ba⟩φ(a) = b, φ(b) = ba
1122293⟨a, b | aaa=1, baaaab=ab⟩φ(a) = b, φ(b) = ba
1122731⟨a, b | aaa=1, aaaab=bab⟩φ(a) = b, φ(b) = ba
1122743⟨a, b | aaa=1, aaabb=aab⟩φ(a) = b, φ(b) = a
1122766⟨a, b | aaa=1, aabba=aba⟩φ(a) = b, φ(b) = a
1122781⟨a, b | aaa=1, abaab=aab⟩φ(a) = b, φ(b) = a
1122825⟨a, b | aaa=1, baaab=aab⟩φ(a) = b, φ(b) = a
1123268⟨a, b | aaa=1, aabb=aaab⟩φ(a) = bb, φ(b) = a
1123274⟨a, b | aaa=1, abab=aaab⟩φ(a) = bb, φ(b) = a
1123280⟨a, b | aaa=1, abba=aaba⟩φ(a) = bb, φ(b) = a
1123291⟨a, b | aaa=1, baaa=aabb⟩φ(a) = bb, φ(b) = a
1123292⟨a, b | aaa=1, baaa=abab⟩φ(a) = bb, φ(b) = a