#10273 ⟨a, b | aa=1, ababa=bbb⟩

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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

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. a2 ⇒ 1 [1]
2. (ba)2 ⇒ ab3 [3]
3. b3a ⇒ (ab)2 [4]
4. b2ab3 ⇒ abab2a [5]
5. bab7 ⇒ aba(b2a)2 [6]
6. b10 ⇒ a(b2a)3 [7]
7. ba(b2a)3 ⇒ a(b2a)3b [8]
# ab:aa=1,ababa=bbb ab - -
aa=1
baba=abbb
bbba=abab
bbabbb=ababba
babbbbbbb=ababbabba
bbbbbbbbbb=abbabbabba
babbabbabba=abbabbabbab

Same cardinality

2 unique, 10 total

Σ#PresentationPropertiesφ
91523⟨a, b | aab=ba, bbbb=1⟩Noc 646
105442⟨a, b | aaaa=1, ababbb=1⟩Grp 642

Isomorphic instances

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

5 total

Σ#PresentationMapping
1010294⟨a, b | aa=1, abbba=bab⟩φ(a) = a, φ(b) = b
1010511⟨a, b | aa=1, baba=abbb⟩φ(a) = a, φ(b) = b
1127113⟨a, b | aa=1, abababa=bb⟩φ(a) = a, φ(b) = ab
1127640⟨a, b | aa=1, ababab=bba⟩φ(a) = a, φ(b) = ab
1128255⟨a, b | aa=1, babab=abba⟩φ(a) = a, φ(b) = ab