#7802 ⟨a, b | aaa=1, abbbb=ba⟩

Properties
Finite structure
Right Cayley graph
Left Cayley graph
Rewriting system
Isomorphic instances
Anti-isomorphic instances

Properties

Sum of relation sides is 10
Finite non-commutative monoid with 192 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b⁶³ = b and b ⋅ 1 = b, however b⁶³ ≠ 1

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) = x^m.

4 element center:
- 1, b²¹, b⁴², b⁶³
1 non-trivial idempotent:
- b⁶³
Order of generators:
- a: order 3
- b: index 1, period 63

Histogram:

order 3	2 elements	a, a²
index 1, period 1	1 element	b⁶³
index 1, period 3	44 elements	ab³, a²b³, ab⁶, a²b⁶, ab⁹, ...
index 1, period 7	6 elements	b⁹, b¹⁸, b²⁷, b³⁶, b⁴⁵, ...
index 1, period 9	90 elements	ab, a²b, ab², a²b², ab⁴, ...
index 1, period 21	12 elements	b³, b⁶, b¹², b¹⁵, b²⁴, ...
index 1, period 63	36 elements	b, b², b⁴, b⁵, b⁸, ...

Right Cayley graph

Left Cayley graph

Rewriting system

Format:	Pretty Plain
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:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b⁶⁴ ⇒ b	[3]
2.	ba ⇒ ab⁴	[2]
3.	a³ ⇒ 1	[1]

# ab:aaa=1,abbbb=ba b/a - -
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
ba=abbbb
aaa=1

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	23323	⟨a, b \| aaa=1, bbbb=aaba⟩	φ(a) = a, φ(b) = b

Anti-isomorphic instances