#7143 ⟨a, b | bb=aa, aaab=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
  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.

1ababb2ab2b3ab3b4ab4b5ab5b6
11ababb2ab2b3ab3b4ab4b5ab5b6
aab2abb3ab2b4ab3b5ab4b6ab5b3ab2
bbab3b2ab4b3ab5b4ab2b5ab3b6ab4b3
ababb5ab2b6ab3b3ab4b4ab5b5ab2b6ab3
b2b2ab2b3ab3b4ab4b5ab5b6ab2b3ab3b4
ab2ab2b4ab3b5ab4b6ab5b3ab2b4ab3b5ab4
b3b3ab5b4ab2b5ab3b6ab4b3ab5b4ab2b5
ab3ab3b3ab4b4ab5b5ab2b6ab3b3ab4b4ab5
b4b4ab4b5ab5b6ab2b3ab3b4ab4b5ab5b6
ab4ab4b6ab5b3ab2b4ab3b5ab4b6ab5b3ab2
b5b5ab3b6ab4b3ab5b4ab2b5ab3b6ab4b3
ab5ab5b5ab2b6ab3b3ab4b4ab5b5ab2b6ab3
b6b6ab2b3ab3b4ab4b5ab5b6ab2b3ab3b4

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. b7 ⇒ b3 [5]
2. ab6 ⇒ ab2 [6]
3. ba ⇒ ab3 [4]
4. a2 ⇒ b2 [1]
# ab:bb=aa,aaab=ba b/a - -
bbbbbbb=bbb
abbbbbb=abb
ba=abbb
aa=bb

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
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.

1 total

Σ#PresentationMapping
107146⟨a, b | bb=aa, aaba=ab⟩φ(a) = b, φ(b) = a