#5300 ⟨a, b | aaa=aa, aba=bb⟩

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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.
The zero element z satisfies zy = yz = z for all y.
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. a3 ⇒ a2 [1]
2. b2 ⇒ aba [2]
3. (ab)2 ⇒ (ba)2 [3]
4. baba2b ⇒ (a2b)2a [4]
5. ab(a2b)2 ⇒ b(a2b)2a [5]
# ab:aaa=aa,aba=bb a/b - -
aaa=aa
bb=aba
abab=baba
babaab=aabaaba
abaabaab=baabaaba

Same cardinality

6 unique, 93 total

Σ#PresentationPropertiesφ
8450⟨a, b | bab=aa, bbb=1⟩Noc 2727
8752⟨a, b | aaa=1, babb=a⟩Grp 2759
91695⟨a, b | bab=aa, bbb=b⟩Noc 271
1115514⟨a, b | aaa=bb, aabaa=b⟩Noc 27
1118739⟨a, b | aaa=a, abbbbb=b⟩Noc 27
1120314⟨a, b | aba=b, bbbb=aaa⟩Noc 27