#20252 ⟨a, b | aba=b, aaaa=bab⟩

Quick links

Properties
Elements
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 24 elements
Not cancellative, because multiplication by a⁴ is not injective:
- a⁴ ⋅ a¹⁰ = a⁴ and a⁴ ⋅ 1 = a⁴, however a¹⁰ ≠ 1

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.

3 element center:
- 1, a⁵, a¹⁰
1 non-trivial idempotent:
- a¹⁰
Order of generators:
- a: index 4, period 10
- b: index 1, period 4

Histogram:

index 1, period 1	1 element	a¹⁰
index 1, period 2	1 element	a⁵
index 1, period 4	10 elements	b, ba, ba², ba³, ba⁴, ...
index 1, period 5	4 elements	a⁴, a⁶, a⁸, a¹²
index 1, period 10	4 elements	a⁷, a⁹, a¹¹, a¹³
index 2, period 5	1 element	a²
index 2, period 10	1 element	a³
index 4, period 10	1 element	a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	a¹²	a¹³
1	1	a	b	a²	ba	a³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	a¹²	a¹³
a	a	a²	ba⁹	a³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a¹²	a¹³	a⁴
b	b	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a¹²	ba⁹	a¹³	b	a⁴	ba	ba²	ba³
a²	a²	a³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	ba⁶	a¹²	ba⁷	a¹³	a⁴	a⁵
ba	ba	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	b	a¹²	ba	a¹³	ba²	ba³	ba⁴
a³	a³	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a¹²	ba⁵	a¹³	ba⁶	a⁴	a⁵	a⁶
ba²	ba²	ba³	a¹³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a¹¹	ba²	a¹²	ba³	ba⁴	ba⁵
a⁴	a⁴	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a¹¹	ba²	a¹²	ba³	a¹³	ba⁴	a⁴	ba⁵	a⁵	a⁶	a⁷
ba³	ba³	ba⁴	a¹²	ba⁵	a¹³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	ba⁵	ba⁶
a⁵	a⁵	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	b	a¹²	ba	a¹³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	a⁷	a⁸
ba⁴	ba⁴	ba⁵	a¹¹	ba⁶	a¹²	ba⁷	a¹³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	ba⁶	ba⁷
a⁶	a⁶	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a¹²	ba⁹	a¹³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	a⁸	a⁹
ba⁵	ba⁵	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a¹²	ba⁹	a¹³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	ba⁷	ba⁸
a⁷	a⁷	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	ba⁶	a¹²	ba⁷	a¹³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	a⁹	a¹⁰
ba⁶	ba⁶	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	b	a¹²	ba	a¹³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	ba⁸	ba⁹
a⁸	a⁸	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a¹²	ba⁵	a¹³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹	a¹⁰	a¹¹
ba⁷	ba⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	ba	a¹¹	ba²	a¹²	ba³	a¹³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	ba⁹	b
a⁹	a⁹	a¹⁰	ba	a¹¹	ba²	a¹²	ba³	a¹³	ba⁴	a⁴	ba⁵	a⁵	ba⁶	a⁶	ba⁷	a⁷	ba⁸	a⁸	ba⁹	a⁹	b	a¹⁰	a¹¹	a¹²
ba⁸	ba⁸	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a¹²	ba⁵	a¹³	ba⁶	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	b	ba
a¹⁰	a¹⁰	a¹¹	b	a¹²	ba	a¹³	ba²	a⁴	ba³	a⁵	ba⁴	a⁶	ba⁵	a⁷	ba⁶	a⁸	ba⁷	a⁹	ba⁸	a¹⁰	ba⁹	a¹¹	a¹²	a¹³
ba⁹	ba⁹	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	ba⁶	a¹²	ba⁷	a¹³	ba⁸	a⁴	ba⁹	a⁵	b	ba	ba²
a¹¹	a¹¹	a¹²	ba⁹	a¹³	b	a⁴	ba	a⁵	ba²	a⁶	ba³	a⁷	ba⁴	a⁸	ba⁵	a⁹	ba⁶	a¹⁰	ba⁷	a¹¹	ba⁸	a¹²	a¹³	a⁴
a¹²	a¹²	a¹³	ba⁸	a⁴	ba⁹	a⁵	b	a⁶	ba	a⁷	ba²	a⁸	ba³	a⁹	ba⁴	a¹⁰	ba⁵	a¹¹	ba⁶	a¹²	ba⁷	a¹³	a⁴	a⁵
a¹³	a¹³	a⁴	ba⁷	a⁵	ba⁸	a⁶	ba⁹	a⁷	b	a⁸	ba	a⁹	ba²	a¹⁰	ba³	a¹¹	ba⁴	a¹²	ba⁵	a¹³	ba⁶	a⁴	a⁵	a⁶

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

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
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a¹⁴ ⇒ a⁴	[8]
2.	ba¹⁰ ⇒ b	[10]
3.	ab ⇒ ba⁹	[9]
4.	b² ⇒ a⁵	[4]

# ab:aba=b,aaaa=bab a/b
aaaaaaaaaaaaaa=aaaa
baaaaaaaaaa=b
ab=baaaaaaaaa
bb=aaaaa

Same cardinality

6 unique, 77 total

Σ	#	Presentation	Description	Related
8	425	⟨a, b \| aab=ba, bbb=1⟩	Finite non-commutative monoid with 24 elements	18 iso, 3 anti-iso
10	5009	⟨a, b \| aba=bb, aabbb=1⟩	Finite non-Abelian group with 24 elements	30 iso
10	6935	⟨a, b \| bb=aa, aaabab=1⟩	Finite non-Abelian group with 24 elements	2 iso
11	11515	⟨a, b \| ababa=b, abbaa=1⟩	Finite non-Abelian group with 24 elements	10 iso
11	13372	⟨a, b \| aaaab=1, bbbbbb=1⟩	Isomorphic to ℤ₂₄	8 iso
11	19638	⟨a, b \| aba=a, aaaab=bb⟩	Finite non-commutative monoid with 24 elements

Up:	Monoids with two generators and two relations
Prev:	#20251 ⟨a, b \| aba=b, aaaa=abb⟩
Next:	#20253 ⟨a, b \| aba=b, aaaa=bbb⟩