#13936 ⟨a, b | aaab=a, bbbbbb=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 78 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.

1 element center:
- 1
6 non-trivial idempotents:
- ba², b²a⁴, b³a⁶, a¹², b⁴a⁸, b⁵a¹⁰
Order of generators:
- a: index 1, period 12
- 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	6 elements	ba², b²a⁴, b³a⁶, a¹², b⁴a⁸, ...
index 1, period 2	6 elements	a⁶, b⁴a², ba⁸, b⁵a⁴, b²a¹⁰, ...
index 1, period 3	12 elements	a⁴, b³a², ba⁶, b⁵a², a⁸, ...
index 1, period 4	12 elements	a³, b²a, ba⁵, b³a³, b⁵a, ...
index 1, period 6	12 elements	a², b²a², ba⁴, b³a⁴, b²a⁶, ...
index 1, period 12	24 elements	a, ba, ba³, b³a, a⁵, ...

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

a¹³ ⇒ a
ab ⇒ a¹¹
b⁶ ⇒ 1

# ab:aaab=a,bbbbbb=1 a/b
aaaaaaaaaaaaa=a
ab=aaaaaaaaaaa
bbbbbb=1

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	14972	⟨a, b \| aaa=ab, bbbbbb=1⟩	φ(a) = ba, φ(b) = b

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