#11787 ⟨a, b | aaab=ba, bbbbb=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 1215 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.

3 element center:
- 1, a¹²¹, a²⁴²
1 non-trivial idempotent:
- a²⁴²
Order of generators:
- a: index 1, period 242
- b: order 5

Histogram:

order 5	4 elements	b, b², b³, b⁴
index 1, period 1	1 element	a²⁴²
index 1, period 2	1 element	a¹²¹
index 1, period 5	484 elements	a²b, a²b², a⁴b, a²b³, a⁴b², ...
index 1, period 10	484 elements	ab, ab², a³b, ab³, a³b², ...
index 1, period 11	10 elements	a²², a⁴⁴, a⁶⁶, a⁸⁸, a¹¹⁰, ...
index 1, period 22	10 elements	a¹¹, a³³, a⁵⁵, a⁷⁷, a⁹⁹, ...
index 1, period 121	110 elements	a², a⁴, a⁶, a⁸, a¹⁰, ...
index 1, period 242	110 elements	a, a³, a⁵, a⁷, a⁹, ...

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:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1

a²⁴³ ⇒ a
ba ⇒ a³b
b⁵ ⇒ 1

# ab:aaab=ba,bbbbb=1 reversed:a/b
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa=a
ba=aaab
bbbbb=1

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#11787 ⟨a, b | aaab=ba, bbbbb=1⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph