#450 ⟨a, b | bab=aa, bbb=1⟩

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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. b3 ⇒ 1
  2. a2 ⇒ bab
  3. (ba)2 ⇒ (ab)2
  4. abab2a ⇒ (b2a)2b
  5. ba(b2a)2 ⇒ a(b2a)2b 
# ab:bab=aa,bbb=1 b/a
bbb=1
aa=bab
baba=abab
ababba=bbabbab
babbabba=abbabbab

Right Cayley graph

Left Cayley graph

Others with same cardinality

6 unique, 66 total

Σ#PresentationDescriptionRelated
8752⟨a, b | aaa=1, babb=a⟩Finite non-Abelian group with 27 elements58 iso, 1 anti-iso
91695⟨a, b | bab=aa, bbb=b⟩Finite non-commutative monoid with 27 elements1 iso
105300⟨a, b | aaa=aa, aba=bb⟩Finite non-commutative monoid with 27 elements
1115514⟨a, b | aaa=bb, aabaa=b⟩Finite non-commutative monoid with 27 elements
1118739⟨a, b | aaa=a, abbbbb=b⟩Finite non-commutative monoid with 27 elements
1120314⟨a, b | aba=b, bbbb=aaa⟩Finite non-commutative monoid with 27 elements

Other isomorphic instances

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

27 total

Σ#PresentationMapping
8747⟨a, b | aaa=1, abba=b⟩φ(a) = bb, φ(b) = a
91305⟨a, b | baa=abb, bbb=1⟩φ(a) = a, φ(b) = bb
91310⟨a, b | bab=aba, bbb=1⟩φ(a) = abb, φ(b) = b
92435⟨a, b | aaa=1, ababa=b⟩φ(a) = bb, φ(b) = ab
92573⟨a, b | aaa=1, aabb=ba⟩φ(a) = b, φ(b) = a
107503⟨a, b | aaa=1, aabbaa=b⟩φ(a) = b, φ(b) = a
107513⟨a, b | aaa=1, abaaba=b⟩φ(a) = bb, φ(b) = abb
107769⟨a, b | aaa=1, aabaa=bb⟩φ(a) = bb, φ(b) = a
107772⟨a, b | aaa=1, aabab=ba⟩φ(a) = b, φ(b) = abb
107775⟨a, b | aaa=1, aabba=ab⟩φ(a) = bb, φ(b) = a
108041⟨a, b | aaa=1, aaba=abb⟩φ(a) = b, φ(b) = a
108058⟨a, b | aaa=1, abab=baa⟩φ(a) = bb, φ(b) = ab
108078⟨a, b | aaa=1, baab=aba⟩φ(a) = b, φ(b) = ab
1121661⟨a, b | aaa=1, aaaabba=b⟩φ(a) = bb, φ(b) = a
1121687⟨a, b | aaa=1, aababaa=b⟩φ(a) = b, φ(b) = abb
1121711⟨a, b | aaa=1, abaaaba=b⟩φ(a) = bb, φ(b) = a
1122202⟨a, b | aaa=1, aaaaba=bb⟩φ(a) = b, φ(b) = a
1122225⟨a, b | aaa=1, aabaab=ba⟩φ(a) = b, φ(b) = ab
1122228⟨a, b | aaa=1, aababa=ab⟩φ(a) = bb, φ(b) = ab
1122744⟨a, b | aaa=1, aaabb=aba⟩φ(a) = b, φ(b) = a
1122754⟨a, b | aaa=1, aabaa=bab⟩φ(a) = bb, φ(b) = ab
1122768⟨a, b | aaa=1, aabba=baa⟩φ(a) = b, φ(b) = a
1122784⟨a, b | aaa=1, abaab=baa⟩φ(a) = bb, φ(b) = abb
1122826⟨a, b | aaa=1, baaab=aba⟩φ(a) = b, φ(b) = a
1123272⟨a, b | aaa=1, abaa=aabb⟩φ(a) = bb, φ(b) = a
1123275⟨a, b | aaa=1, abab=aaba⟩φ(a) = b, φ(b) = abb
1123279⟨a, b | aaa=1, abba=aaab⟩φ(a) = bb, φ(b) = a