#1907 ⟨a, b | aab=a, bbbbb=1⟩

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Contents

  1. Properties
  2. Finite structure
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. Same cardinality
  7. Isomorphic instances
  8. 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.

Right Cayley graph

Left Cayley graph

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 ⇒ a [7]
2. ab ⇒ a5 [6]
3. b5 ⇒ 1 [2]
# ab:aab=a,bbbbb=1 a/b - -
aaaaaa=a
ab=aaaaa
bbbbb=1

Same cardinality

8 unique, 108 total

Σ#PresentationPropertiesφ
91470⟨a, b | aaa=bb, abab=1⟩Grp 3073
92443⟨a, b | aaa=1, abbbb=b⟩Noc 3019
106560⟨a, b | aaa=a, bbbb=ab⟩Noc 30
107013⟨a, b | ab=aa, bbbbb=b⟩Noc 30
1111058⟨a, b | abba=bbb, baba=1⟩Grp 308
1113115⟨a, b | bbb=aaa, aaba=b⟩Noc 30
1115979⟨a, b | aaa=aa, bbbb=ab⟩Noc 30
1120847⟨a, b | ab=aa, bbbbb=bb⟩Noc 30

Isomorphic instances

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

11 total

Σ#PresentationMapping
92155⟨a, b | ab=aa, bbbbb=1⟩φ(a) = bba, φ(b) = b
104541⟨a, b | aabb=a, bbbbb=1⟩φ(a) = aa, φ(b) = b
104573⟨a, b | abab=a, bbbbb=1⟩φ(a) = aa, φ(b) = b
105001⟨a, b | aba=ab, bbbbb=1⟩φ(a) = ba, φ(b) = b
1111448⟨a, b | aabbb=a, bbbbb=1⟩φ(a) = a, φ(b) = bb
1111533⟨a, b | ababb=a, bbbbb=1⟩φ(a) = a, φ(b) = bb
1111549⟨a, b | abbab=a, bbbbb=1⟩φ(a) = a, φ(b) = bb
1112096⟨a, b | abba=ab, aaaaa=1⟩φ(a) = b, φ(b) = a
1112127⟨a, b | abba=ab, bbbbb=1⟩φ(a) = a, φ(b) = b
1112772⟨a, b | abb=aba, aaaaa=1⟩φ(a) = b, φ(b) = bba
1112803⟨a, b | abb=aba, bbbbb=1⟩φ(a) = bba, φ(b) = b

Anti-isomorphic instances

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

4 total

Σ#PresentationMapping
104918⟨a, b | aab=bb, aaaaa=1⟩φ(a) = bb, φ(b) = a
1111788⟨a, b | aaab=bb, aaaaa=1⟩φ(a) = b, φ(b) = aa
1111980⟨a, b | aabb=ab, aaaaa=1⟩φ(a) = b, φ(b) = a
1112756⟨a, b | abb=aab, aaaaa=1⟩φ(a) = bbbb, φ(b) = a