#2259 ⟨a, b | ab=aa, bbb=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

Properties

Finite structure

Elements in the center commute with all other elements.
A left zero element x satisfies xy = x for all y.
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.

1aba2bab2a3ba2b2aba3b2a2b2a3
11aba2bab2a3ba2b2aba3b2a2b2a3
aaa2a2a3a3a3a3a3a3a3a3a3
bbbab2ba2b2ab2ba3b2a2b2ab2a3b2a2b2a3
a2a2a3a3a3a3a3a3a3a3a3a3a3
bababa2ba2ba3ba3ba3ba3ba3ba3ba3ba3ba3
b2b2b2ab2b2a2b2ab2b2a3b2a2b2ab2a3b2a2b2a3
a3a3a3a3a3a3a3a3a3a3a3a3a3
ba2ba2ba3ba3ba3ba3ba3ba3ba3ba3ba3ba3ba3
b2ab2ab2a2b2a2b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3
ba3ba3ba3ba3ba3ba3ba3ba3ba3ba3ba3ba3ba3
b2a2b2a2b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3
b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3b2a3

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. ab ⇒ a2 [1]
2. b3 ⇒ b2 [2]
3. a4 ⇒ a3 [3]
# ab:ab=aa,bbb=bb ab - -
ab=aa
bbb=bb
aaaa=aaa

Same cardinality

37 unique, 659 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
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