#13218 ⟨a, b | abb=aba, bbb=aa⟩

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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.
The zero element z satisfies zy = yz = z 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.

1aba2abbab2a3a2babababb2aa4a2ba(ba)2b2abb(ba)2
11aba2abbab2a3a2babababb2aa4a2ba(ba)2b2abb(ba)2
aaa2aba3a2babaabaa4a4a2baa3a4a4a4a4a4a4
bbbab2a2bbabb2aa2a2baa2ba(ba)2b2aba3a4a4b(ba)2a4a4
a2a2a3a2ba4a4a2baa2baa4a4a4a4a4a4a4a4a4a4
abababaabaa4a3a4a3a4a4a4a4a4a4a4a4a4a4
babaa2bbaba2baa2ba(ba)2(ba)2a4a4a4a2baa4a4a4a4a4a4
b2b2b2aa2a2bab2aba3a2ba4a4b(ba)2a4a2baa4a4a4a4a4
a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
a2ba2ba2baa2baa4a4a4a4a4a4a4a4a4a4a4a4a4a4
abaabaa4a3a4a4a4a4a4a4a4a4a4a4a4a4a4a4
babbab(ba)2(ba)2a4a2baa4a2baa4a4a4a4a4a4a4a4a4a4
b2ab2aa2bab2aba4a4b(ba)2b(ba)2a4a4a4a4a4a4a4a4a4a4
a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
a2baa2baa4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
(ba)2(ba)2a4a2baa4a4a4a4a4a4a4a4a4a4a4a4a4a4
b2abb2abb(ba)2b(ba)2a4a4a4a4a4a4a4a4a4a4a4a4a4a4
b(ba)2b(ba)2a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4

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. ab2 ⇒ aba [1]
2. ba2 ⇒ a2b [3]
3. b3 ⇒ a2 [2]
4. a3b ⇒ a4 [6]
5. (ab)2 ⇒ a3 [4]
6. a5 ⇒ a4 [7]
# ab:abb=aba,bbb=aa ab - -
abb=aba
baa=aab
bbb=aa
aaab=aaaa
abab=aaa
aaaaa=aaaa

Same cardinality

16 unique, 71 total

Σ#PresentationPropertiesDescriptionφ
104375⟨a, b | aaaa=b, abbbb=1⟩Grp Com 17ℤ1739
106428⟨a, b | aab=b, babbb=a⟩Noc 172
106514⟨a, b | aba=b, baaab=a⟩Noc 17
107152⟨a, b | bb=aa, abab=aa⟩Noc 17
107153⟨a, b | bb=aa, abab=ab⟩Noc 17
107154⟨a, b | bb=aa, abab=ba⟩Noc 17
1113141⟨a, b | aab=aaa, bbb=ab⟩Noc 17
1113142⟨a, b | aab=aaa, bbb=ba⟩Noc 17
1113151⟨a, b | aba=aaa, baa=bb⟩Noc 17
1113156⟨a, b | aba=aaa, bbb=ab⟩Noc 17
1114395⟨a, b | aaaa=b, abbbb=a⟩Noc Com 17ℕ(17 = 1)2
1114396⟨a, b | aaaa=b, abbbb=b⟩Noc Com 17ℕ(17 = 4)2
1114698⟨a, b | aabb=a, baaaa=b⟩Noc 177
1115488⟨a, b | aaa=ab, babbb=b⟩Noc 173
1116143⟨a, b | aab=aa, bbbb=ba⟩Noc 17
1116521⟨a, b | aab=bb, bab=aaa⟩Noc 17