#15514 ⟨a, b | aaa=bb, aabaa=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 27 elements

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

5 element center:
- 1, a³, a⁶, a⁹, a¹²
1 non-trivial idempotent:
- a¹²
Order of generators:
- a: index 3, period 12
- b: index 1, period 8

Histogram:

index 1, period 1	1 element	a¹²
index 1, period 2	1 element	a⁶
index 1, period 3	2 elements	a⁴, a⁸
index 1, period 4	2 elements	a³, a⁹
index 1, period 6	2 elements	a¹⁰, a¹⁴
index 1, period 8	12 elements	b, ba, ba², ba³, ba⁴, ...
index 1, period 12	4 elements	a⁵, a⁷, a¹¹, a¹³
index 2, period 6	1 element	a²
index 3, period 12	1 element	a

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

a¹⁵ ⇒ a³
ba¹² ⇒ b
ab ⇒ ba⁵
b² ⇒ a³

# ab:aaa=bb,aabaa=b a/b
aaaaaaaaaaaaaaa=aaa
baaaaaaaaaaaa=b
ab=baaaaa
bb=aaa

Right Cayley graph

Left Cayley graph

Others with same cardinality

6 unique, 93 total

Σ	#	Presentation	Description	Related
8	450	⟨a, b \| bab=aa, bbb=1⟩	Finite non-commutative monoid with 27 elements	27 iso
8	752	⟨a, b \| aaa=1, babb=a⟩	Finite non-Abelian group with 27 elements	58 iso, 1 anti-iso
9	1695	⟨a, b \| bab=aa, bbb=b⟩	Finite non-commutative monoid with 27 elements	1 iso
10	5300	⟨a, b \| aaa=aa, aba=bb⟩	Finite non-commutative monoid with 27 elements
11	18739	⟨a, b \| aaa=a, abbbbb=b⟩	Finite non-commutative monoid with 27 elements
11	20314	⟨a, b \| aba=b, bbbb=aaa⟩	Finite non-commutative monoid with 27 elements

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