#4749 ⟨a, b | abba=b, baab=a⟩

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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. a7 ⇒ a [7]
2. ba6 ⇒ b [8]
3. a3b ⇒ ba3 [3]
4. b2 ⇒ a2ba2 [6]
5. bab ⇒ aba [5]
6. ba2b ⇒ a [2]
# ab:abba=b,baab=a a/b - -
aaaaaaa=a
baaaaaa=b
aaab=baaa
bb=aabaa
bab=aba
baab=a

Same cardinality

7 unique, 13 total

Σ#PresentationPropertiesDescriptionφ
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
1114342⟨a, b | aaaa=a, aabba=b⟩Noc 25
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