#7013 ⟨a, b | ab=aa, bbbbb=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. ab ⇒ a2 [1]
2. b5 ⇒ b [2]
3. a6 ⇒ a2 [3]
# ab:ab=aa,bbbbb=b ab - -
ab=aa
bbbbb=b
aaaaaa=aa

Same cardinality

8 unique, 123 total

Σ#PresentationPropertiesφ
91470⟨a, b | aaa=bb, abab=1⟩Grp 3073
91907⟨a, b | aab=a, bbbbb=1⟩Noc 3015
92443⟨a, b | aaa=1, abbbb=b⟩Noc 3019
106560⟨a, b | aaa=a, bbbb=ab⟩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