#4095 ⟨a, b | baa=abb, abab=1⟩

Properties

Presentation has sum-of-sides 10
Finite non-Abelian group with 20 elements
Inverses of generators:
- a^-1 = a³
- b^-1 = aba

Element profile

1 element center:
- 1

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 shortlex with a < b

ba² ⇒ ab²
bab ⇒ a³
b³ ⇒ aba
a⁴ ⇒ 1

# ab:baa=abb,abab=1 ab
baa=abb
bab=aaa
bbb=aba
aaaa=1

Cayley table

	1	a	b	a²	ab	ba	b²	a³	a²b	aba	ab²	b²a	a³b	a²ba	a²b²	ab²a	a³ba	a³b²	a²b²a	a³b²a
1	1	a	b	a²	ab	ba	b²	a³	a²b	aba	ab²	b²a	a³b	a²ba	a²b²	ab²a	a³ba	a³b²	a²b²a	a³b²a
a	a	a²	ab	a³	a²b	aba	ab²	1	a³b	a²ba	a²b²	ab²a	b	a³ba	a³b²	a²b²a	ba	b²	a³b²a	b²a
b	b	ba	b²	ab²	a³	b²a	aba	ab²a	a²ba	1	a³b	a²b²	a²b²a	a³b²	a	a³ba	ab	a³b²a	a²	a²b
a²	a²	a³	a²b	1	a³b	a²ba	a²b²	a	b	a³ba	a³b²	a²b²a	ab	ba	b²	a³b²a	aba	ab²	b²a	ab²a
ab	ab	aba	ab²	a²b²	1	ab²a	a²ba	a²b²a	a³ba	a	b	a³b²	a³b²a	b²	a²	ba	a²b	b²a	a³	a³b
ba	ba	ab²	a³	ab²a	a²ba	1	a³b	b	a²b²a	a³b²	a	a³ba	b²	ab	a³b²a	a²	b²a	aba	a²b	a²b²
b²	b²	b²a	aba	a³b	ab²a	a²b²	1	a³ba	a³b²	b	a²b²a	a	a²	a³b²a	ba	ab	a³	a²b	ab²	a²ba
a³	a³	1	a³b	a	b	a³ba	a³b²	a²	ab	ba	b²	a³b²a	a²b	aba	ab²	b²a	a²ba	a²b²	ab²a	a²b²a
a²b	a²b	a²ba	a²b²	a³b²	a	a²b²a	a³ba	a³b²a	ba	a²	ab	b²	b²a	ab²	a³	aba	a³b	ab²a	1	b
aba	aba	a²b²	1	a²b²a	a³ba	a	b	ab	a³b²a	b²	a²	ba	ab²	a²b	b²a	a³	ab²a	a²ba	a³b	a³b²
ab²	ab²	ab²a	a²ba	b	a²b²a	a³b²	a	ba	b²	ab	a³b²a	a²	a³	b²a	aba	a²b	1	a³b	a²b²	a³ba
b²a	b²a	a³b	ab²a	a³ba	a³b²	b	a²b²a	b²	a²	a³b²a	ba	ab	aba	a³	a²b	ab²	a²b²	1	a²ba	a
a³b	a³b	a³ba	a³b²	b²	a²	a³b²a	ba	b²a	aba	a³	a²b	ab²	ab²a	a²b²	1	a²ba	b	a²b²a	a	ab
a²ba	a²ba	a³b²	a	a³b²a	ba	a²	ab	a²b	b²a	ab²	a³	aba	a²b²	a³b	ab²a	1	a²b²a	a³ba	b	b²
a²b²	a²b²	a²b²a	a³ba	ab	a³b²a	b²	a²	aba	ab²	a²b	b²a	a³	1	ab²a	a²ba	a³b	a	b	a³b²	ba
ab²a	ab²a	b	a²b²a	ba	b²	ab	a³b²a	ab²	a³	b²a	aba	a²b	a²ba	1	a³b	a²b²	a³b²	a	a³ba	a²
a³ba	a³ba	b²	a²	b²a	aba	a³	a²b	a³b	ab²a	a²b²	1	a²ba	a³b²	b	a²b²a	a	a³b²a	ba	ab	ab²
a³b²	a³b²	a³b²a	ba	a²b	b²a	ab²	a³	a²ba	a²b²	a³b	ab²a	1	a	a²b²a	a³ba	b	a²	ab	b²	aba
a²b²a	a²b²a	ab	a³b²a	aba	ab²	a²b	b²a	a²b²	1	ab²a	a²ba	a³b	a³ba	a	b	a³b²	b²	a²	ba	a³
a³b²a	a³b²a	a²b	b²a	a²ba	a²b²	a³b	ab²a	a³b²	a	a²b²a	a³ba	b	ba	a²	ab	b²	ab²	a³	aba	1

Right Cayley graph

Left Cayley graph

Others with same cardinality

20 unique, 60 total

Σ	#	Presentation	Description	Related
8	526	⟨a, b \| aab=a, bbbb=1⟩	Finite non-commutative monoid with 20 elements	10 iso, 9 anti-iso
9	2207	⟨a, b \| ab=aa, bbbb=b⟩	Finite non-commutative monoid with 20 elements	1 anti-iso
10	4212	⟨a, b \| aaaaa=1, abbbb=1⟩	Isomorphic to ℤ₂₀	16 iso
10	5349	⟨a, b \| aaa=bb, bab=aa⟩	Finite non-commutative monoid with 20 elements	1 iso
10	6728	⟨a, b \| aba=a, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
10	6764	⟨a, b \| aba=b, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
10	7117	⟨a, b \| ab=aa, bbbb=bb⟩	Finite non-commutative monoid with 20 elements
11	12159	⟨a, b \| aaaa=aa, abbb=b⟩	Finite non-commutative monoid with 20 elements
11	12322	⟨a, b \| aaab=bb, bbba=a⟩	Finite non-commutative monoid with 20 elements
11	14367	⟨a, b \| aaaa=a, bbbbb=a⟩	Isomorphic to ℕ_{(20 = 5)}
11	14407	⟨a, b \| aaaa=b, bbbbb=a⟩	Isomorphic to ℕ_{(20 = 1)}
11	14408	⟨a, b \| aaaa=b, bbbbb=b⟩	Isomorphic to ℕ_{(20 = 4)}
11	15933	⟨a, b \| aba=bb, baaab=a⟩	Finite non-commutative monoid with 20 elements
11	16124	⟨a, b \| aab=aa, baba=bb⟩	Finite non-commutative monoid with 20 elements
11	16339	⟨a, b \| aba=aa, aaaa=bb⟩	Finite non-commutative monoid with 20 elements
11	16343	⟨a, b \| aba=aa, aaab=bb⟩	Finite non-commutative monoid with 20 elements
11	16545	⟨a, b \| aba=bb, abb=aaa⟩	Finite non-commutative monoid with 20 elements
11	18619	⟨a, b \| aba=b, aaaaabb=1⟩	Finite non-Abelian group with 20 elements	3 iso
11	19503	⟨a, b \| aab=a, bbbbb=bb⟩	Finite non-commutative monoid with 20 elements
11	21047	⟨a, b \| ab=aa, bbbb=bbb⟩	Finite non-commutative monoid with 20 elements

Other isomorphic instances

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

4 total

Σ	#	Presentation	Mapping
10	5915	⟨a, b \| abab=1, abbaa=b⟩	φ(a) = a, φ(b) = b
11	15164	⟨a, b \| aab=ba, abbabb=1⟩	φ(a) = aaab, φ(b) = a
11	15182	⟨a, b \| aab=ba, babbab=1⟩	φ(a) = aaab, φ(b) = a
11	15191	⟨a, b \| aab=ba, bbabba=1⟩	φ(a) = aaab, φ(b) = a

Other anti-isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
10	5907	⟨a, b \| abab=1, aabba=b⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
Prev:	#4089 ⟨a, b \| baa=aab, bbbb=1⟩
Next:	#4107 ⟨a, b \| bab=aaa, bbbb=1⟩