#19638 ⟨a, b | aba=a, aaaab=bb⟩

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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 ⋅ a⁵ = a⁴ and a⁴ ⋅ 1 = a⁴, however a⁵ ≠ 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.

1 element center:
- 1
4 non-trivial idempotents:
- ab, ba, a⁵, a⁶b
Order of generators:
- a: index 4, period 5
- b: index 2, period 5

Histogram:

index 1, period 1	4 elements	ab, ba, a⁵, a⁶b
index 1, period 5	8 elements	a⁴, a⁴b, a⁶, a⁵b, a⁷, ...
index 2, period 1	1 element	ba²b
index 2, period 5	7 elements	b, a², a³, bab, a³b, ...
index 3, period 5	2 elements	a²b, ba²
index 4, period 5	1 element	a

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	a³b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
1	1	a	b	a²	ab	ba	a³	a²b	ba²	bab	a⁴	a³b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a	a	a²	ab	a³	a²b	a	a⁴	a³b	a²	ab	a⁵	a⁴b	a³	a²b	a⁶	a⁵b	a³b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
b	b	ba	a⁴b	ba²	bab	a⁴	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a²	a²	a³	a²b	a⁴	a³b	a²	a⁵	a⁴b	a³	a²b	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ab	ab	a	a⁵b	a²	ab	a⁵	a³	a²b	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
ba	ba	ba²	bab	ba³	ba²b	ba	a⁸	ba³b	ba²	bab	a⁴	a⁸b	ba³	ba²b	a⁵	a⁴b	ba³b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a³	a³	a⁴	a³b	a⁵	a⁴b	a³	a⁶	a⁵b	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a²b	a²b	a²	a⁶b	a³	a²b	a⁶	a⁴	a³b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
ba²	ba²	ba³	ba²b	a⁸	ba³b	ba²	a⁴	a⁸b	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
bab	bab	ba	a⁴b	ba²	bab	a⁴	ba³	ba²b	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a⁴	a⁴	a⁵	a⁴b	a⁶	a⁵b	a⁴	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a³b	a³b	a³	a⁷b	a⁴	a³b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ba³	ba³	a⁸	ba³b	a⁴	a⁸b	ba³	a⁵	a⁴b	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
ba²b	ba²b	ba²	a⁵b	ba³	ba²b	a⁵	a⁸	ba³b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁵	a⁵	a⁶	a⁵b	a⁷	a⁶b	a⁵	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁴b	a⁴b	a⁴	a⁸b	a⁵	a⁴b	a⁸	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
ba³b	ba³b	ba³	a⁶b	a⁸	ba³b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁶	a⁶	a⁷	a⁶b	a⁸	a⁷b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁵b	a⁵b	a⁵	a⁴b	a⁶	a⁵b	a⁴	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁶b	a⁵	a⁴b	a⁶	a⁵b	a⁷	a⁶b	a⁷b
a⁷	a⁷	a⁸	a⁷b	a⁴	a⁸b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b
a⁶b	a⁶b	a⁶	a⁵b	a⁷	a⁶b	a⁵	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁷b	a⁶	a⁵b	a⁷	a⁶b	a⁸	a⁷b	a⁸b
a⁸	a⁸	a⁴	a⁸b	a⁵	a⁴b	a⁸	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁵b	a⁴	a⁸b	a⁵	a⁴b	a⁶	a⁵b	a⁶b
a⁷b	a⁷b	a⁷	a⁶b	a⁸	a⁷b	a⁶	a⁴	a⁸b	a⁷	a⁶b	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁸b	a⁷	a⁶b	a⁸	a⁷b	a⁴	a⁸b	a⁴b
a⁸b	a⁸b	a⁸	a⁷b	a⁴	a⁸b	a⁷	a⁵	a⁴b	a⁸	a⁷b	a⁶	a⁵b	a⁴	a⁸b	a⁷	a⁶b	a⁴b	a⁸	a⁷b	a⁴	a⁸b	a⁵	a⁴b	a⁵b

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.	a⁹ ⇒ a⁴	[5]
2.	ba⁴ ⇒ a⁸	[4]
3.	aba ⇒ a	[1]
4.	b² ⇒ a⁴b	[2]

# ab:aba=a,aaaab=bb a/b
aaaaaaaaa=aaaa
baaaa=aaaaaaaa
aba=a
bb=aaaab

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	20252	⟨a, b \| aba=b, aaaa=bab⟩	Finite non-commutative monoid with 24 elements

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