#15200 ⟨a, b | aab=ba, bbbbbb=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 384 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.

2 element center:
- 1, a⁶³
1 non-trivial idempotent:
- a⁶³
Order of generators:
- a: index 1, period 63
- b: order 6

Histogram:

order 2	1 element	b³
order 3	2 elements	b², b⁴
order 6	2 elements	b, b⁵
index 1, period 1	1 element	a⁶³
index 1, period 2	9 elements	a⁷b³, a¹⁴b³, a²¹b³, a²⁸b³, a³⁵b³, ...
index 1, period 3	44 elements	a³b², a³b⁴, a⁶b², a⁶b⁴, a⁹b², ...
index 1, period 6	126 elements	ab, a²b, a³b, a⁴b, a⁵b, ...
index 1, period 7	6 elements	a⁹, a¹⁸, a²⁷, a³⁶, a⁴⁵, ...
index 1, period 9	90 elements	ab², a²b², ab⁴, a⁴b², a²b⁴, ...
index 1, period 14	54 elements	ab³, a²b³, a³b³, a⁴b³, a⁵b³, ...
index 1, period 21	12 elements	a³, a⁶, a¹², a¹⁵, a²⁴, ...
index 1, period 63	36 elements	a, a², a⁴, a⁵, 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:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1

a⁶⁴ ⇒ a
ba ⇒ a²b
b⁶ ⇒ 1

# ab:aab=ba,bbbbbb=1 reversed:a/b
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa=a
ba=aab
bbbbbb=1

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#15200 ⟨a, b | aab=ba, bbbbbb=1⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph