#3385 ⟨a, b | aa=1, abbbba=b⟩

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Properties

Elements

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.

Complete rewriting system

Format:
Word to reduce:
Tips:
  • Lowercase letters stand for generators.
  • Spaces are ignored.
  • Numbers repeat the previous letter, e.g. b90.
Reduction strategy:
Path to normal form: 1 
1
  1. b16 ⇒ b
  2. ba ⇒ ab4
  3. a2 ⇒ 1 
# ab:aa=1,abbbba=b b/a
bbbbbbbbbbbbbbbb=b
ba=abbbb
aa=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

2 unique, 2 total

Σ#PresentationDescriptionRelated
105421⟨a, b | aba=bb, bab=aa⟩Finite non-commutative monoid with 32 elements
1115515⟨a, b | aaa=bb, aabab=a⟩Finite non-commutative monoid with 32 elements

Other isomorphic instances

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

17 total

Σ#PresentationMapping
93522⟨a, b | aa=1, abbbb=ba⟩φ(a) = a, φ(b) = b
93676⟨a, b | aa=1, bbbb=aba⟩φ(a) = a, φ(b) = b
1126499⟨a, b | aa=1, aaabbbba=b⟩φ(a) = a, φ(b) = b
1126577⟨a, b | aa=1, abaabbba=b⟩φ(a) = a, φ(b) = b
1126609⟨a, b | aa=1, abbaabba=b⟩φ(a) = a, φ(b) = b
1127028⟨a, b | aa=1, aaabbbb=ba⟩φ(a) = a, φ(b) = b
1127077⟨a, b | aa=1, aabbbba=ab⟩φ(a) = a, φ(b) = b
1127105⟨a, b | aa=1, abaabbb=ba⟩φ(a) = a, φ(b) = b
1127136⟨a, b | aa=1, abbaabb=ba⟩φ(a) = a, φ(b) = b
1127151⟨a, b | aa=1, abbbaab=ba⟩φ(a) = a, φ(b) = b
1127606⟨a, b | aa=1, aabbbb=aba⟩φ(a) = a, φ(b) = b
1127683⟨a, b | aa=1, abbbba=aab⟩φ(a) = a, φ(b) = b
1127720⟨a, b | aa=1, baabbb=aba⟩φ(a) = a, φ(b) = b
1127750⟨a, b | aa=1, bbaabb=aba⟩φ(a) = a, φ(b) = b
1128065⟨a, b | aa=1, aaaba=bbbb⟩φ(a) = a, φ(b) = b
1128210⟨a, b | aa=1, abbbb=aaba⟩φ(a) = a, φ(b) = b
1128216⟨a, b | aa=1, abbbb=baaa⟩φ(a) = a, φ(b) = b