#5334 ⟨a, b | aaa=ab, bba=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.

1aba2bab2a3ba2a4ba3a5ba4ba5
11aba2bab2a3ba2a4ba3a5ba4ba5
aaa2a3a3a4a5a4a5a5a5a5a5a5
bbbab2ba2b2b2ba3b2ba4b2ba5b2b2
a2a2a3a4a4a5a5a5a5a5a5a5a5a5
bababa2ba3ba3ba4ba5ba4ba5ba5ba5ba5ba5ba5
b2b2b2b2b2b2b2b2b2b2b2b2b2b2
a3a3a4a5a5a5a5a5a5a5a5a5a5a5
ba2ba2ba3ba4ba4ba5ba5ba5ba5ba5ba5ba5ba5ba5
a4a4a5a5a5a5a5a5a5a5a5a5a5a5
ba3ba3ba4ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5
a5a5a5a5a5a5a5a5a5a5a5a5a5a5
ba4ba4ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5
ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5ba5

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. a6 ⇒ a5 [3]
2. ab ⇒ a3 [1]
3. b2a ⇒ b2 [2]
4. b3 ⇒ b2 [4]
# ab:aaa=ab,bba=bb a/b - -
aaaaaa=aaaaa
ab=aaa
bba=bb
bbb=bb

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
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
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