#16459 ⟨a, b | aba=bb, abbb=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
  8. Isomorphic instances

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.

1aba2abbaa3abababa4(ba)2a5a6
11aba2abbaa3abababa4(ba)2a5a6
aaa2aba3a4abaa4a5(ba)2a5a2a6a2
bbbaabaa4baba5a5(ba)2a2a6a3a2a3
a2a2a3a4a4a5a5a5a6a2a6a3a2a3
abababaa5a5(ba)2a6a6a2a3a2a4a3a4
babaa4baba5a6(ba)2a6a2a3a2a4a3a4
a3a3a4a5a5a6a6a6a2a3a2a4a3a4
abaabaa5(ba)2a6a2a2a2a3a4a3a5a4a5
babbab(ba)2a2a2a3a3a3a4a5a4a6a5a6
a4a4a5a6a6a2a2a2a3a4a3a5a4a5
(ba)2(ba)2a2a3a3a4a4a4a5a6a5a2a6a2
a5a5a6a2a2a3a3a3a4a5a4a6a5a6
a6a6a2a3a3a4a4a4a5a6a5a2a6a2

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 ⇒ a2 [7]
2. ba2 ⇒ a4 [9]
3. a2b ⇒ a4 [6]
4. b2 ⇒ aba [1]
5. (ab)2 ⇒ (ba)2 [3]
# ab:aba=bb,abbb=aa a/b - -
aaaaaaa=aa
baa=aaaa
aab=aaaa
bb=aba
abab=baba

Same cardinality

26 unique, 302 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
105340⟨a, b | aaa=bb, aab=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
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

Isomorphic instances

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

1 total

Σ#PresentationMapping
1116466⟨a, b | aba=bb, babb=aa⟩φ(a) = a, φ(b) = b