#6101 ⟨a, b | aab=a, bbbbbb=1⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 42 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:
- ba, b²a², a⁶, b³a³, b⁴a⁴, b⁵a⁵
Order of generators:
- a: index 1, period 6
- 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	ba, b²a², a⁶, b³a³, b⁴a⁴, ...
index 1, period 2	6 elements	a³, ba⁴, b⁴a, b²a⁵, b⁵a², ...
index 1, period 3	12 elements	a², a⁴, ba³, b³a, ba⁵, ...
index 1, period 6	12 elements	a, ba², b²a, a⁵, b²a³, ...

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

a⁷ ⇒ a
ab ⇒ a⁶
b⁶ ⇒ 1

# ab:aab=a,bbbbbb=1 a/b
aaaaaaa=a
ab=aaaaaa
bbbbbb=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

2 unique, 9 total

Σ	#	Presentation	Description	Related
11	14981	⟨a, b \| aaa=bb, aabaab=1⟩	Finite non-Abelian group with 42 elements	7 iso
11	20647	⟨a, b \| ab=aa, bbbbbb=b⟩	Finite non-commutative monoid with 42 elements

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

3 total

Σ	#	Presentation	Mapping
10	6909	⟨a, b \| ab=aa, bbbbbb=1⟩	φ(a) = bba, φ(b) = b
11	14256	⟨a, b \| abab=a, bbbbbb=1⟩	φ(a) = aa, φ(b) = b
11	15364	⟨a, b \| aba=ab, bbbbbb=1⟩	φ(a) = ba, φ(b) = b

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