#12157 ⟨a, b | aaaa=aa, abba=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

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.

1aba2abb2a3ab2b3ab3
11aba2abb2a3ab2b3ab3
aaa2aba3bab2a2b2ab3b3
bbab2b2bab3b3ab2abbab2
a2a2a3ba2abb2a3ab2b3ab3
ababb2ab2abb3ab3b2babb2
b2b2abb3b2ab2babab3b2ab
a3a3a2aba3bab2a2b2ab3b3
ab2ab2bab3ab2b2abbb3ab2b
b3b3ab3bb3abb2ab3ab2b3ab3
ab3ab3b3abab3bab2b3b2ab3b3

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. b4 ⇒ b [12]
2. ba ⇒ ab2 [9]
3. a2b ⇒ b [6]
4. a4 ⇒ a2 [1]
# ab:aaaa=aa,abba=b b/a - -
bbbb=b
ba=abb
aab=b
aaaa=aa

Same cardinality

31 unique, 919 total

Σ#PresentationPropertiesDescriptionφ
7332⟨a, b | aa=1, abbb=b⟩Noc 1085
8507⟨a, b | aaa=b, abbb=1⟩Grp Com 10ℤ10536
8656⟨a, b | bb=aa, aba=a⟩Noc 101
8657⟨a, b | bb=aa, aba=b⟩Noc 101
8970⟨a, b | aa=a, bbb=ab⟩Noc 103
91654⟨a, b | aab=bb, bba=a⟩Noc 106
92012⟨a, b | aaa=b, abbb=a⟩Noc Com 10ℕ(10 = 1)59
92013⟨a, b | aaa=b, abbb=b⟩Noc Com 10ℕ(10 = 3)35
92277⟨a, b | bb=aa, aba=aa⟩Noc 101
92894⟨a, b | aa=a, bbbbb=a⟩Noc Com 10ℕ(10 = 5)17
92935⟨a, b | aa=b, bbbbb=b⟩Noc Com 10ℕ(10 = 2)57
104637⟨a, b | aaaa=b, aabb=b⟩Noc Com 10ℕ(10 = 4)20
105346⟨a, b | aaa=bb, abb=ab⟩Noc 103
105356⟨a, b | aab=aa, aba=bb⟩Noc 10
106587⟨a, b | aaa=b, abbb=bb⟩Noc Com 10ℕ(10 = 6)3
106633⟨a, b | aab=a, baaa=bb⟩Noc 1021
107089⟨a, b | ab=aa, baaa=bb⟩Noc 103
108619⟨a, b | aa=a, abbbbb=b⟩Noc 105
109051⟨a, b | ab=a, baaaa=bb⟩Noc 104
1110729⟨a, b | aaab=baa, abab=1⟩Grp 103
1112181⟨a, b | aaaa=ab, baaa=b⟩Noc 101
1112196⟨a, b | aaaa=bb, aaab=a⟩Noc Com 107
1112197⟨a, b | aaaa=bb, aaab=b⟩Noc Com 101
1112452⟨a, b | aabb=aa, bbbb=a⟩Noc Com 10ℕ(10 = 8)3
1115738⟨a, b | aab=bb, aaaaa=b⟩Noc Com 10ℕ(10 = 7)7
1119564⟨a, b | aab=b, abbbb=aa⟩Noc Com 105
1119899⟨a, b | aaa=b, abbb=bbb⟩Noc Com 10ℕ(10 = 9)1
1120052⟨a, b | aab=b, aaaa=baa⟩Noc 10
1120253⟨a, b | aba=b, aaaa=bbb⟩Noc Com 10
1125905⟨a, b | ab=a, bbbb=baaa⟩Noc 10
1125909⟨a, b | ab=a, bbbb=bbaa⟩Noc 10