#402 ⟨a, b | aaa=ab, bbb=1⟩

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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
  9. Anti-isomorphic instances

Properties

Finite structure

Elements in the center commute with all other elements.
An idempotent element x satisfies x2 = x.
The order of x is the least n (if it exists) such that xn = 1.
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.

1aba2bab2a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4ba6b2a5b2a6
11aba2bab2a3ba2b2aa4ba3b2a2a5ba4b2a3a6ba5b2a4ba6b2a5b2a6
aaa2a3a3a4a5a4a5a6a5a6aa6aa2aa2a3a3a4a5
bbbab2ba2b2a1ba3b2a2aba4b2a3a2ba5b2a4a3ba6b2a5a4b2a6a5a6
a2a2a3a4a4a5a6a5a6aa6aa2aa2a3a2a3a4a4a5a6
bababa2ba3ba3ba4ba5ba4ba5ba6ba5ba6baba6baba2baba2ba3ba3ba4ba5
b2b2b2a1b2a2abb2a3a2bab2a4a3ba2b2a5a4ba3b2a6a5ba4a6ba5ba6
a3a3a4a5a5a6aa6aa2aa2a3a2a3a4a3a4a5a5a6a
ba2ba2ba3ba4ba4ba5ba6ba5ba6baba6baba2baba2ba3ba2ba3ba4ba4ba5ba6
b2ab2ab2a2b2a3b2a3b2a4b2a5b2a4b2a5b2a6b2a5b2a6b2ab2a6b2ab2a2b2ab2a2b2a3b2a3b2a4b2a5
a4a4a5a6a6aa2aa2a3a2a3a4a3a4a5a4a5a6a6aa2
ba3ba3ba4ba5ba5ba6baba6baba2baba2ba3ba2ba3ba4ba3ba4ba5ba5ba6ba
b2a2b2a2b2a3b2a4b2a4b2a5b2a6b2a5b2a6b2ab2a6b2ab2a2b2ab2a2b2a3b2a2b2a3b2a4b2a4b2a5b2a6
a5a5a6aaa2a3a2a3a4a3a4a5a4a5a6a5a6aaa2a3
ba4ba4ba5ba6ba6baba2baba2ba3ba2ba3ba4ba3ba4ba5ba4ba5ba6ba6baba2
b2a3b2a3b2a4b2a5b2a5b2a6b2ab2a6b2ab2a2b2ab2a2b2a3b2a2b2a3b2a4b2a3b2a4b2a5b2a5b2a6b2a
a6a6aa2a2a3a4a3a4a5a4a5a6a5a6aa6aa2a2a3a4
ba5ba5ba6bababa2ba3ba2ba3ba4ba3ba4ba5ba4ba5ba6ba5ba6bababa2ba3
b2a4b2a4b2a5b2a6b2a6b2ab2a2b2ab2a2b2a3b2a2b2a3b2a4b2a3b2a4b2a5b2a4b2a5b2a6b2a6b2ab2a2
ba6ba6baba2ba2ba3ba4ba3ba4ba5ba4ba5ba6ba5ba6baba6baba2ba2ba3ba4
b2a5b2a5b2a6b2ab2ab2a2b2a3b2a2b2a3b2a4b2a3b2a4b2a5b2a4b2a5b2a6b2a5b2a6b2ab2ab2a2b2a3
b2a6b2a6b2ab2a2b2a2b2a3b2a4b2a3b2a4b2a5b2a4b2a5b2a6b2a5b2a6b2ab2a6b2ab2a2b2a2b2a3b2a4

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 ⇒ a [3]
2. ab ⇒ a3 [1]
3. b3 ⇒ 1 [2]
# ab:aaa=ab,bbb=1 a/b - -
aaaaaaa=a
ab=aaa
bbb=1

Same cardinality

8 unique, 25 total

Σ#PresentationPropertiesDescriptionφ
91599⟨a, b | aaa=ab, bbb=b⟩Noc 211
105309⟨a, b | aaa=aa, bbb=ab⟩Noc 21
106265⟨a, b | aaa=a, abbbb=b⟩Noc 212
1111127⟨a, b | aaaaa=b, abbbb=1⟩Grp Com 21ℤ2113
1113091⟨a, b | bab=aaa, abba=b⟩Noc 21
1113248⟨a, b | bab=aaa, bbb=aa⟩Noc 21
1115158⟨a, b | aab=ba, ababab=1⟩Grp 211
1119212⟨a, b | aba=b, baaaab=a⟩Noc 21

Isomorphic instances

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

12 total

Σ#PresentationMapping
91292⟨a, b | abb=aaa, bbb=1⟩φ(a) = a, φ(b) = bb
107798⟨a, b | aaa=1, abbba=ab⟩φ(a) = bb, φ(b) = a
107809⟨a, b | aaa=1, baabb=ba⟩φ(a) = bb, φ(b) = a
108070⟨a, b | aaa=1, abbb=aba⟩φ(a) = b, φ(b) = a
108086⟨a, b | aaa=1, babb=baa⟩φ(a) = b, φ(b) = a
1121677⟨a, b | aaa=1, aaabbba=b⟩φ(a) = bb, φ(b) = a
1122221⟨a, b | aaa=1, aaabbb=ba⟩φ(a) = b, φ(b) = a
1122266⟨a, b | aaa=1, ababba=ab⟩φ(a) = b, φ(b) = a
1122297⟨a, b | aaa=1, baaabb=ba⟩φ(a) = b, φ(b) = a
1122741⟨a, b | aaa=1, aaaba=bbb⟩φ(a) = b, φ(b) = a
1122804⟨a, b | aaa=1, abbab=aba⟩φ(a) = b, φ(b) = ba
1123287⟨a, b | aaa=1, abbb=abaa⟩φ(a) = bb, φ(b) = a

Anti-isomorphic instances

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

27 total

Σ#PresentationMapping
8749⟨a, b | aaa=1, abbb=b⟩φ(a) = bb, φ(b) = a
92431⟨a, b | aaa=1, aabbb=b⟩φ(a) = b, φ(b) = a
92439⟨a, b | aaa=1, abbab=b⟩φ(a) = b, φ(b) = a
92590⟨a, b | aaa=1, babb=ab⟩φ(a) = b, φ(b) = ba
107501⟨a, b | aaa=1, aababb=b⟩φ(a) = b, φ(b) = ba
107515⟨a, b | aaa=1, abaabb=b⟩φ(a) = b, φ(b) = ba
107517⟨a, b | aaa=1, ababab=b⟩φ(a) = b, φ(b) = ba
107779⟨a, b | aaa=1, aabbb=ab⟩φ(a) = bb, φ(b) = a
107812⟨a, b | aaa=1, babab=ab⟩φ(a) = bb, φ(b) = a
108069⟨a, b | aaa=1, abbb=aab⟩φ(a) = b, φ(b) = a
1121663⟨a, b | aaa=1, aaaabbb=b⟩φ(a) = bb, φ(b) = a
1121689⟨a, b | aaa=1, aababab=b⟩φ(a) = bb, φ(b) = a
1121695⟨a, b | aaa=1, aabbaab=b⟩φ(a) = bb, φ(b) = a
1121713⟨a, b | aaa=1, abaaabb=b⟩φ(a) = bb, φ(b) = a
1121721⟨a, b | aaa=1, ababaab=b⟩φ(a) = bb, φ(b) = a
1121733⟨a, b | aaa=1, abbaaab=b⟩φ(a) = bb, φ(b) = a
1122220⟨a, b | aaa=1, aaabbb=ab⟩φ(a) = b, φ(b) = a
1122239⟨a, b | aaa=1, aabbab=ab⟩φ(a) = b, φ(b) = a
1122244⟨a, b | aaa=1, aabbba=ba⟩φ(a) = b, φ(b) = a
1122296⟨a, b | aaa=1, baaabb=ab⟩φ(a) = b, φ(b) = a
1122300⟨a, b | aaa=1, baabab=ab⟩φ(a) = b, φ(b) = a
1122733⟨a, b | aaa=1, aaaab=bbb⟩φ(a) = b, φ(b) = a
1122795⟨a, b | aaa=1, ababb=aab⟩φ(a) = b, φ(b) = ba
1122831⟨a, b | aaa=1, baabb=aab⟩φ(a) = b, φ(b) = ba
1122839⟨a, b | aaa=1, babab=aab⟩φ(a) = b, φ(b) = ba
1123284⟨a, b | aaa=1, abbb=aaab⟩φ(a) = bb, φ(b) = a
1123293⟨a, b | aaa=1, baaa=abbb⟩φ(a) = bb, φ(b) = a