#14631 ⟨a, b | aaba=b, abbbb=a⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 61 elements

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
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, a⁵, a¹⁰, a¹⁵
1 non-trivial idempotent:
- a¹⁵
Order of generators:
- a: index 1, period 15
- b: index 1, period 4

Histogram:

index 1, period 1	1 element	a¹⁵
index 1, period 2	5 elements	b², b²a³, b²a⁶, b²a⁹, b²a¹²
index 1, period 3	2 elements	a⁵, a¹⁰
index 1, period 4	10 elements	b, b³, ba³, b³a³, ba⁶, ...
index 1, period 5	4 elements	a³, a⁶, a⁹, a¹²
index 1, period 6	10 elements	b²a, b²a², b²a⁴, b²a⁵, b²a⁷, ...
index 1, period 12	20 elements	ba, ba², b³a, ba⁴, b³a², ...
index 1, period 15	8 elements	a, a², a⁴, a⁷, a⁸, ...

Complete rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(a) = 0; deg(b) = 1

a¹⁶ ⇒ a
ba¹⁵ ⇒ b
ab ⇒ ba⁷
b⁴ ⇒ a¹⁵

# ab:aaba=b,abbbb=a a/b
aaaaaaaaaaaaaaaa=a
baaaaaaaaaaaaaaa=b
ab=baaaaaaa
bbbb=aaaaaaaaaaaaaaa

Right Cayley graph

Left Cayley graph

Other isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	14661	⟨a, b \| aaba=b, bbbba=a⟩	φ(a) = a, φ(b) = b

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