#5340 ⟨a, b | aaa=bb, aab=ba⟩

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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.

1aba2aba3a2ba4a3ba5a4ba6a5b
11aba2aba3a2ba4a3ba5a4ba6a5b
aaa2aba3a2ba4a3ba5a4ba6a5ba4a3b
bba2ba3a4ba5a3ba4a5ba6a4ba5a3ba4
a2a2a3a2ba4a3ba5a4ba6a5ba4a3ba5a4b
ababa3ba4a5ba6a4ba5a3ba4a5ba6a4ba5
a3a3a4a3ba5a4ba6a5ba4a3ba5a4ba6a5b
a2ba2ba4ba5a3ba4a5ba6a4ba5a3ba4a5ba6
a4a4a5a4ba6a5ba4a3ba5a4ba6a5ba4a3b
a3ba3ba5ba6a4ba5a3ba4a5ba6a4ba5a3ba4
a5a5a6a5ba4a3ba5a4ba6a5ba4a3ba5a4b
a4ba4ba3ba4a5ba6a4ba5a3ba4a5ba6a4ba5
a6a6a4a3ba5a4ba6a5ba4a3ba5a4ba6a5b
a5ba5ba4ba5a3ba4a5ba6a4ba5a3ba4a5ba6

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. a7 ⇒ a4 [4]
2. a6b ⇒ a3b [3]
3. ba ⇒ a2b [2]
4. b2 ⇒ a3 [1]
# ab:aaa=bb,aab=ba reversed:a/b - -
aaaaaaa=aaaa
aaaaaab=aaab
ba=aab
bb=aaa

Same cardinality

26 unique, 303 total

Σ#PresentationPropertiesDescriptionφ
91328⟨a, b | aaaa=b, abbb=1⟩Grp Com 13ℤ13189
92118⟨a, b | aba=b, baab=a⟩Noc 134
104642⟨a, b | aaaa=b, abbb=a⟩Noc Com 13ℕ(13 = 1)10
104643⟨a, b | aaaa=b, abbb=b⟩Noc Com 13ℕ(13 = 4)6
104683⟨a, b | aaab=b, abba=a⟩Noc 1316
105065⟨a, b | aaa=ab, babb=b⟩Noc 137
105334⟨a, b | aaa=ab, bba=bb⟩Noc 13
105336⟨a, b | aaa=ab, bbb=ab⟩Noc 131
105337⟨a, b | aaa=ab, bbb=ba⟩Noc 13
106305⟨a, b | aaa=b, abbbb=b⟩Noc Com 13ℕ(13 = 3)2
107143⟨a, b | bb=aa, aaab=ba⟩Noc 131
1112240⟨a, b | aaab=aa, bbbb=a⟩Noc Com 13ℕ(13 = 8)1
1112268⟨a, b | aaab=ab, bbbb=a⟩Noc Com 13ℕ(13 = 5)3
1112324⟨a, b | aaab=bb, bbbb=a⟩Noc Com 13ℕ(13 = 2)14
1114647⟨a, b | aaba=b, babbb=a⟩Noc Com 132
1115520⟨a, b | aaa=bb, aabbb=b⟩Noc Com 132
1116012⟨a, b | aaa=ab, abbb=bb⟩Noc 13
1116069⟨a, b | aaa=bb, abbb=ba⟩Noc 131
1116205⟨a, b | aab=ab, bbbb=aa⟩Noc 132
1116459⟨a, b | aba=bb, abbb=aa⟩Noc 131
1116506⟨a, b | aab=ab, bbb=aaa⟩Noc 131
1116515⟨a, b | aab=ba, bbb=aaa⟩Noc 13
1118958⟨a, b | aab=a, bbbbbb=a⟩Noc Com 13ℕ(13 = 6)4
1120046⟨a, b | aab=a, bbbb=bba⟩Noc 13
1120927⟨a, b | ab=aa, aaaa=bbb⟩Noc 137
1120991⟨a, b | ab=aa, baaa=bbb⟩Noc 133