#1434 ⟨a, b | baab=a, bbbb=1⟩

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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. a13 ⇒ a [13]
2. aba12 ⇒ ab [12]
3. a3b ⇒ ba3 [3]
4. ba2b ⇒ a [1]
5. ab2 ⇒ baba2 [7]
6. a(ab)2 ⇒ b2a [6]
7. b3a ⇒ a2b [5]
8. b(ab)2 ⇒ aba10 [10]
9. b4 ⇒ 1 [2]
# ab:baab=a,bbbb=1 a/b - -
aaaaaaaaaaaaa=a
abaaaaaaaaaaaa=ab
aaab=baaa
baab=a
abb=babaa
aabab=bba
bbba=aab
babab=abaaaaaaaaaa
bbbb=1

Same cardinality

1 unique, 3 total

Σ#PresentationPropertiesφ
1113293⟨a, b | aaaaa=1, ababbb=1⟩Grp 1002

Isomorphic instances

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

9 total

Σ#PresentationMapping
91566⟨a, b | aba=bb, aaaa=1⟩φ(a) = bbb, φ(b) = a
104108⟨a, b | bab=aba, aaaa=1⟩φ(a) = bbb, φ(b) = ab
105713⟨a, b | aaaa=1, ababa=b⟩φ(a) = b, φ(b) = aab
1111051⟨a, b | abba=bab, bbbb=1⟩φ(a) = ababa, φ(b) = b
1111066⟨a, b | abbb=baa, bbbb=1⟩φ(a) = a, φ(b) = b
1111089⟨a, b | babb=aab, bbbb=1⟩φ(a) = a, φ(b) = bbb
1117121⟨a, b | aaaa=1, abaaba=b⟩φ(a) = b, φ(b) = aa
1117629⟨a, b | aaaa=1, aaabb=ba⟩φ(a) = bbb, φ(b) = a
1117639⟨a, b | aaaa=1, aabba=ab⟩φ(a) = b, φ(b) = a