#14342 ⟨a, b | aaaa=a, aabba=b⟩

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Contents

  1. Properties
  2. Finite structure
  3. Right Cayley graph
  4. Left Cayley graph
  5. Rewriting system
  6. 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.

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. b8 ⇒ b [12]
2. ab ⇒ b2a [10]
3. ba3 ⇒ b [5]
4. a4 ⇒ a [1]
# ab:aaaa=a,aabba=b reversed:b/a - -
bbbbbbbb=b
ab=bba
baaa=b
aaaa=a

Same cardinality

7 unique, 13 total

Σ#PresentationPropertiesDescriptionφ
104749⟨a, b | abba=b, baab=a⟩Noc 25
106381⟨a, b | aab=a, bbbbb=b⟩Noc 251
106536⟨a, b | aaa=a, aaba=bb⟩Noc 25
1111133⟨a, b | aaaaa=b, bbbbb=1⟩Grp Com 25ℤ253
1114869⟨a, b | abba=b, bbabb=a⟩Noc 252
1119634⟨a, b | aba=a, aaaaa=bb⟩Noc 25
1119706⟨a, b | aba=b, aaaaa=bb⟩Noc 25