#425 ⟨a, b | aab=ba, bbb=1⟩

Properties
Finite structure
Cayley table
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Isomorphic instances
Anti-isomorphic instances

Properties

Sum of relation sides is 8
Finite non-commutative monoid with 24 elements
Not cancellative, because multiplication by a is not injective:
- a ⋅ a⁷ = a and a ⋅ 1 = a, however a⁷ ≠ 1

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.

2 element center:
- 1, a⁷
1 non-trivial idempotent:
- a⁷
Order of generators:
- a: index 1, period 7
- b: order 3
Histogram:

order 3 2 elements b, b²

index 1, period 1 1 element a⁷

index 1, period 3 14 elements ab, a²b, ab², a³b, a²b², ...

index 1, period 7 6 elements a, a², a³, a⁴, a⁵, ...

Cayley table

Idempotents are shown in bold.

	1	a	b	a²	ab	b²	a³	a²b	ab²	a⁴	a³b	a²b²	a⁵	a⁴b	a³b²	a⁶	a⁵b	a⁴b²	a⁷	a⁶b	a⁵b²	a⁷b	a⁶b²	a⁷b²
1	1	a	b	a²	ab	b²	a³	a²b	ab²	a⁴	a³b	a²b²	a⁵	a⁴b	a³b²	a⁶	a⁵b	a⁴b²	a⁷	a⁶b	a⁵b²	a⁷b	a⁶b²	a⁷b²
a	a	a²	ab	a³	a²b	ab²	a⁴	a³b	a²b²	a⁵	a⁴b	a³b²	a⁶	a⁵b	a⁴b²	a⁷	a⁶b	a⁵b²	a	a⁷b	a⁶b²	ab	a⁷b²	ab²
b	b	a²b	b²	a⁴b	a²b²	1	a⁶b	a⁴b²	a²	ab	a⁶b²	a⁴	a³b	ab²	a⁶	a⁵b	a³b²	a	a⁷b	a⁵b²	a³	a⁷b²	a⁵	a⁷
a²	a²	a³	a²b	a⁴	a³b	a²b²	a⁵	a⁴b	a³b²	a⁶	a⁵b	a⁴b²	a⁷	a⁶b	a⁵b²	a	a⁷b	a⁶b²	a²	ab	a⁷b²	a²b	ab²	a²b²
ab	ab	a³b	ab²	a⁵b	a³b²	a	a⁷b	a⁵b²	a³	a²b	a⁷b²	a⁵	a⁴b	a²b²	a⁷	a⁶b	a⁴b²	a²	ab	a⁶b²	a⁴	ab²	a⁶	a
b²	b²	a⁴b²	1	ab²	a⁴	b	a⁵b²	a	a⁴b	a²b²	a⁵	ab	a⁶b²	a²	a⁵b	a³b²	a⁶	a²b	a⁷b²	a³	a⁶b	a⁷	a³b	a⁷b
a³	a³	a⁴	a³b	a⁵	a⁴b	a³b²	a⁶	a⁵b	a⁴b²	a⁷	a⁶b	a⁵b²	a	a⁷b	a⁶b²	a²	ab	a⁷b²	a³	a²b	ab²	a³b	a²b²	a³b²
a²b	a²b	a⁴b	a²b²	a⁶b	a⁴b²	a²	ab	a⁶b²	a⁴	a³b	ab²	a⁶	a⁵b	a³b²	a	a⁷b	a⁵b²	a³	a²b	a⁷b²	a⁵	a²b²	a⁷	a²
ab²	ab²	a⁵b²	a	a²b²	a⁵	ab	a⁶b²	a²	a⁵b	a³b²	a⁶	a²b	a⁷b²	a³	a⁶b	a⁴b²	a⁷	a³b	ab²	a⁴	a⁷b	a	a⁴b	ab
a⁴	a⁴	a⁵	a⁴b	a⁶	a⁵b	a⁴b²	a⁷	a⁶b	a⁵b²	a	a⁷b	a⁶b²	a²	ab	a⁷b²	a³	a²b	ab²	a⁴	a³b	a²b²	a⁴b	a³b²	a⁴b²
a³b	a³b	a⁵b	a³b²	a⁷b	a⁵b²	a³	a²b	a⁷b²	a⁵	a⁴b	a²b²	a⁷	a⁶b	a⁴b²	a²	ab	a⁶b²	a⁴	a³b	ab²	a⁶	a³b²	a	a³
a²b²	a²b²	a⁶b²	a²	a³b²	a⁶	a²b	a⁷b²	a³	a⁶b	a⁴b²	a⁷	a³b	ab²	a⁴	a⁷b	a⁵b²	a	a⁴b	a²b²	a⁵	ab	a²	a⁵b	a²b
a⁵	a⁵	a⁶	a⁵b	a⁷	a⁶b	a⁵b²	a	a⁷b	a⁶b²	a²	ab	a⁷b²	a³	a²b	ab²	a⁴	a³b	a²b²	a⁵	a⁴b	a³b²	a⁵b	a⁴b²	a⁵b²
a⁴b	a⁴b	a⁶b	a⁴b²	ab	a⁶b²	a⁴	a³b	ab²	a⁶	a⁵b	a³b²	a	a⁷b	a⁵b²	a³	a²b	a⁷b²	a⁵	a⁴b	a²b²	a⁷	a⁴b²	a²	a⁴
a³b²	a³b²	a⁷b²	a³	a⁴b²	a⁷	a³b	ab²	a⁴	a⁷b	a⁵b²	a	a⁴b	a²b²	a⁵	ab	a⁶b²	a²	a⁵b	a³b²	a⁶	a²b	a³	a⁶b	a³b
a⁶	a⁶	a⁷	a⁶b	a	a⁷b	a⁶b²	a²	ab	a⁷b²	a³	a²b	ab²	a⁴	a³b	a²b²	a⁵	a⁴b	a³b²	a⁶	a⁵b	a⁴b²	a⁶b	a⁵b²	a⁶b²
a⁵b	a⁵b	a⁷b	a⁵b²	a²b	a⁷b²	a⁵	a⁴b	a²b²	a⁷	a⁶b	a⁴b²	a²	ab	a⁶b²	a⁴	a³b	ab²	a⁶	a⁵b	a³b²	a	a⁵b²	a³	a⁵
a⁴b²	a⁴b²	ab²	a⁴	a⁵b²	a	a⁴b	a²b²	a⁵	ab	a⁶b²	a²	a⁵b	a³b²	a⁶	a²b	a⁷b²	a³	a⁶b	a⁴b²	a⁷	a³b	a⁴	a⁷b	a⁴b
a⁷	a⁷	a	a⁷b	a²	ab	a⁷b²	a³	a²b	ab²	a⁴	a³b	a²b²	a⁵	a⁴b	a³b²	a⁶	a⁵b	a⁴b²	a⁷	a⁶b	a⁵b²	a⁷b	a⁶b²	a⁷b²
a⁶b	a⁶b	ab	a⁶b²	a³b	ab²	a⁶	a⁵b	a³b²	a	a⁷b	a⁵b²	a³	a²b	a⁷b²	a⁵	a⁴b	a²b²	a⁷	a⁶b	a⁴b²	a²	a⁶b²	a⁴	a⁶
a⁵b²	a⁵b²	a²b²	a⁵	a⁶b²	a²	a⁵b	a³b²	a⁶	a²b	a⁷b²	a³	a⁶b	a⁴b²	a⁷	a³b	ab²	a⁴	a⁷b	a⁵b²	a	a⁴b	a⁵	ab	a⁵b
a⁷b	a⁷b	a²b	a⁷b²	a⁴b	a²b²	a⁷	a⁶b	a⁴b²	a²	ab	a⁶b²	a⁴	a³b	ab²	a⁶	a⁵b	a³b²	a	a⁷b	a⁵b²	a³	a⁷b²	a⁵	a⁷
a⁶b²	a⁶b²	a³b²	a⁶	a⁷b²	a³	a⁶b	a⁴b²	a⁷	a³b	ab²	a⁴	a⁷b	a⁵b²	a	a⁴b	a²b²	a⁵	ab	a⁶b²	a²	a⁵b	a⁶	a²b	a⁶b
a⁷b²	a⁷b²	a⁴b²	a⁷	ab²	a⁴	a⁷b	a⁵b²	a	a⁴b	a²b²	a⁵	ab	a⁶b²	a²	a⁵b	a³b²	a⁶	a²b	a⁷b²	a³	a⁶b	a⁷	a³b	a⁷b

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a⁸ ⇒ a	[3]
2.	ba ⇒ a²b	[1]
3.	b³ ⇒ 1	[2]

# ab:aab=ba,bbb=1 reversed:a/b - -
aaaaaaaa=a
ba=aab
bbb=1

Same cardinality

6 unique, 56 total

Σ	#	Presentation	Properties	Description	φ
10	5009	⟨a, b \| aba=bb, aabbb=1⟩	Grp 24		30
10	6935	⟨a, b \| bb=aa, aaabab=1⟩	Grp 24		2
11	11515	⟨a, b \| ababa=b, abbaa=1⟩	Grp 24		10
11	13372	⟨a, b \| aaaab=1, bbbbbb=1⟩	Grp Com 24	ℤ₂₄	8
11	19638	⟨a, b \| aba=a, aaaab=bb⟩	Noc 24
11	20252	⟨a, b \| aba=b, aaaa=bab⟩	Noc 24

Isomorphic instances

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

12 total

Σ	#	Presentation	Mapping
9	2429	⟨a, b \| aaa=1, aabba=b⟩	φ(a) = b, φ(b) = a
9	2577	⟨a, b \| aaa=1, abab=ba⟩	φ(a) = bb, φ(b) = ab
10	7784	⟨a, b \| aaa=1, abaab=ba⟩	φ(a) = bb, φ(b) = abb
10	8043	⟨a, b \| aaa=1, aaba=bab⟩	φ(a) = bb, φ(b) = ab
10	8050	⟨a, b \| aaa=1, aabb=baa⟩	φ(a) = b, φ(b) = a
10	8063	⟨a, b \| aaa=1, abba=aab⟩	φ(a) = b, φ(b) = a
11	21683	⟨a, b \| aaa=1, aabaaba=b⟩	φ(a) = b, φ(b) = ab
11	22216	⟨a, b \| aaa=1, aaabba=ab⟩	φ(a) = b, φ(b) = a
11	22732	⟨a, b \| aaa=1, aaaab=bba⟩	φ(a) = b, φ(b) = a
11	22753	⟨a, b \| aaa=1, aabaa=abb⟩	φ(a) = b, φ(b) = a
11	22758	⟨a, b \| aaa=1, aabab=aba⟩	φ(a) = bb, φ(b) = ab
11	23296	⟨a, b \| aaa=1, baab=aaba⟩	φ(a) = bb, φ(b) = abb

Anti-isomorphic instances

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

9 total

Σ	#	Presentation	Mapping
9	2570	⟨a, b \| aaa=1, aaba=bb⟩	φ(a) = b, φ(b) = a
10	7499	⟨a, b \| aaa=1, aababa=b⟩	φ(a) = bb, φ(b) = ab
10	8048	⟨a, b \| aaa=1, aabb=aba⟩	φ(a) = b, φ(b) = a
11	22205	⟨a, b \| aaa=1, aaaabb=ba⟩	φ(a) = b, φ(b) = a
11	22236	⟨a, b \| aaa=1, aabbaa=ab⟩	φ(a) = b, φ(b) = a
11	22252	⟨a, b \| aaa=1, abaaab=ba⟩	φ(a) = b, φ(b) = a
11	22737	⟨a, b \| aaa=1, aaaba=abb⟩	φ(a) = b, φ(b) = a
11	22760	⟨a, b \| aaa=1, aabab=baa⟩	φ(a) = bb, φ(b) = ab
11	22789	⟨a, b \| aaa=1, ababa=aab⟩	φ(a) = bb, φ(b) = ab

Up:	Monoid enumeration
Prev:	#413 ⟨a, b \| aab=aa, bbb=1⟩
Next:	#435 ⟨a, b \| aba=aa, bbb=1⟩

order 3	2 elements	b, b²
index 1, period 1	1 element	a⁷
index 1, period 3	14 elements	ab, a²b, ab², a³b, a²b², ...
index 1, period 7	6 elements	a, a², a³, a⁴, a⁵, ...

#425 ⟨a, b | aab=ba, bbb=1⟩

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