#3818 ⟨a, b | aaab=ba, bbbb=1⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 324 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, a⁴⁰, a⁸⁰
1 non-trivial idempotent:
- a⁸⁰
Order of generators:
- a: index 1, period 80
- b: order 4

Histogram:

order 2	1 element	b²
order 4	2 elements	b, b³
index 1, period 1	1 element	a⁸⁰
index 1, period 2	11 elements	a⁸b², a¹⁶b², a²⁴b², a³²b², a⁴⁰, ...
index 1, period 4	92 elements	a²b, a⁴b, a²b³, a⁴b², a⁶b, ...
index 1, period 5	4 elements	a¹⁶, a³², a⁴⁸, a⁶⁴
index 1, period 8	104 elements	ab, a³b, a²b², ab³, a⁵b, ...
index 1, period 10	4 elements	a⁸, a²⁴, a⁵⁶, a⁷²
index 1, period 16	48 elements	ab², a⁵, a³b², a⁵b², a⁷b², ...
index 1, period 20	8 elements	a⁴, a¹², a²⁸, a³⁶, a⁴⁴, ...
index 1, period 40	16 elements	a², a⁶, a¹⁴, a¹⁸, a²², ...
index 1, period 80	32 elements	a, a³, a⁷, a⁹, a¹¹, ...

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

a⁸¹ ⇒ a
ba ⇒ a³b
b⁴ ⇒ 1

# ab:aaab=ba,bbbb=1 reversed:a/b
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa=a
ba=aaab
bbbb=1

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#3818 ⟨a, b | aaab=ba, bbbb=1⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph