#22290 ⟨a, b | aaa=1, abbbbb=ba⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 375 elements

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) = x^m.

5 element center:
- 1, b³¹, b⁶², b⁹³, b¹²⁴
1 non-trivial idempotent:
- b¹²⁴
Order of generators:
- a: order 3
- b: index 1, period 124

Histogram:

order 3	2 elements	a, a²
index 1, period 1	1 element	b¹²⁴
index 1, period 2	1 element	b⁶²
index 1, period 3	62 elements	ab⁴, a²b⁴, ab⁸, a²b⁸, ab¹², ...
index 1, period 4	2 elements	b³¹, b⁹³
index 1, period 6	62 elements	ab², a²b², ab⁶, a²b⁶, ab¹⁰, ...
index 1, period 12	124 elements	ab, a²b, ab³, a²b³, ab⁵, ...
index 1, period 31	30 elements	b⁴, b⁸, b¹², b¹⁶, b²⁰, ...
index 1, period 62	30 elements	b², b⁶, b¹⁰, b¹⁴, b¹⁸, ...
index 1, period 124	60 elements	b, b³, b⁵, b⁷, b⁹, ...

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(b) = 0; deg(a) = 1

b¹²⁵ ⇒ b
ba ⇒ ab⁵
a³ ⇒ 1

# ab:aaa=1,abbbbb=ba b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
ba=abbbbb
aaa=1

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#22290 ⟨a, b | aaa=1, abbbbb=ba⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph