#5065 ⟨a, b | aaa=ab, babb=b⟩

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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.

1aba2abb2ab2b3ab3b4ab4b5ab5
11aba2abb2ab2b3ab3b4ab4b5ab5
aaa2ababab4ab2ab5ab3abab4ab2ab5ab3
bbb4b2b2b5b3bb4b2b5b3bb4
a2a2abab4ab4ab2ab5ab3abab4ab2ab5ab3ab
ababab4ab2ab2ab5ab3abab4ab2ab5ab3abab4
b2b2b5b3b3bb4b2b5b3bb4b2b5
ab2ab2ab5ab3ab3abab4ab2ab5ab3abab4ab2ab5
b3b3bb4b4b2b5b3bb4b2b5b3b
ab3ab3abab4ab4ab2ab5ab3abab4ab2ab5ab3ab
b4b4b2b5b5b3bb4b2b5b3bb4b2
ab4ab4ab2ab5ab5ab3abab4ab2ab5ab3abab4ab2
b5b5b3bbb4b2b5b3bb4b2b5b3
ab5ab5ab3ababab4ab2ab5ab3abab4ab2ab5ab3

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. b6 ⇒ b [9]
2. ba ⇒ b4 [8]
3. a2b ⇒ ab4 [6]
4. a3 ⇒ ab [1]
# ab:aaa=ab,babb=b b/a - -
bbbbbb=b
ba=bbbb
aab=abbbb
aaa=ab

Same cardinality

26 unique, 296 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
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
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.

7 total

Σ#PresentationMapping
105069⟨a, b | aaa=ab, bbab=b⟩φ(a) = a, φ(b) = b
105071⟨a, b | aaa=ab, bbba=b⟩φ(a) = a, φ(b) = b
1115476⟨a, b | aaa=ab, baaab=b⟩φ(a) = a, φ(b) = b
1115478⟨a, b | aaa=ab, baaba=b⟩φ(a) = a, φ(b) = b
1115482⟨a, b | aaa=ab, babaa=b⟩φ(a) = a, φ(b) = b
1115490⟨a, b | aaa=ab, bbaaa=b⟩φ(a) = a, φ(b) = b
1119991⟨a, b | aab=a, baaa=bbb⟩φ(a) = bb, φ(b) = a