#16515 ⟨a, b | aab=ba, bbb=aaa⟩

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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.

1aba2abb2a3a2bab2a4a3ba2b2a3b2
11aba2abb2a3a2bab2a4a3ba2b2a3b2
aaa2aba3a2bab2a4a3ba2b2a4a3ba3b2a3b2
bba2bb2a3ba2b2a3a3ba3b2a4a3ba3b2a4a4
a2a2a3a2ba4a3ba2b2a4a3ba3b2a4a3ba3b2a3b2
ababa3bab2a3ba3b2a4a3ba3b2a4a3ba3b2a4a4
b2b2a3b2a3a3b2a4a3ba3b2a4a3ba3b2a4a3ba3b
a3a3a4a3ba4a3ba3b2a4a3ba3b2a4a3ba3b2a3b2
a2ba2ba3ba2b2a3ba3b2a4a3ba3b2a4a3ba3b2a4a4
ab2ab2a3b2a4a3b2a4a3ba3b2a4a3ba3b2a4a3ba3b
a4a4a4a3ba4a3ba3b2a4a3ba3b2a4a3ba3b2a3b2
a3ba3ba3ba3b2a3ba3b2a4a3ba3b2a4a3ba3b2a4a4
a2b2a2b2a3b2a4a3b2a4a3ba3b2a4a3ba3b2a4a3ba3b
a3b2a3b2a3b2a4a3b2a4a3ba3b2a4a3ba3b2a4a3ba3b

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. a5 ⇒ a4 [10]
2. a4b ⇒ a3b [9]
3. ba ⇒ a2b [1]
4. b3 ⇒ a3 [2]
# ab:aab=ba,bbb=aaa reversed:a/b - -
aaaaa=aaaa
aaaab=aaab
ba=aab
bbb=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
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
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