#7815 ⟨a, b | aaa=1, babbb=ab⟩

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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. b10 ⇒ b
  2. bab ⇒ ab8
  3. ba2b ⇒ a2b5
  4. a3 ⇒ 1 
# ab:aaa=1,babbb=ab b/a
bbbbbbbbbb=b
bab=abbbbbbbb
baab=aabbbbb
aaa=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 22 total

Σ#PresentationDescriptionRelated
92450⟨a, b | aaa=1, babbb=a⟩Finite non-Abelian group with 84 elements13 iso, 8 anti-iso

Other isomorphic instances

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

7 total

Σ#PresentationMapping
107819⟨a, b | aaa=1, bbabb=ab⟩φ(a) = aa, φ(b) = b
1121693⟨a, b | aaa=1, aababbb=b⟩φ(a) = a, φ(b) = b
1121699⟨a, b | aaa=1, aabbabb=b⟩φ(a) = aa, φ(b) = b
1121719⟨a, b | aaa=1, abaabbb=b⟩φ(a) = aa, φ(b) = b
1121727⟨a, b | aaa=1, ababbab=b⟩φ(a) = a, φ(b) = b
1121735⟨a, b | aaa=1, abbaabb=b⟩φ(a) = a, φ(b) = b
1121737⟨a, b | aaa=1, abbabab=b⟩φ(a) = aa, φ(b) = b