#18619 ⟨a, b | aba=b, aaaaabb=1⟩

Properties

Presentation has sum-of-sides 11
Finite non-Abelian group with 20 elements
Inverses of generators:
- a^-1 = a⁹
- b^-1 = ba⁵

Element profile

2 element center:
- 1, a⁵

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

a¹⁰ ⇒ 1
ab ⇒ ba⁹
b² ⇒ a⁵

# ab:aba=b,aaaaabb=1 a/b
aaaaaaaaaa=1
ab=baaaaaaaaa
bb=aaaaa

Cayley table

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

Right Cayley graph

Left Cayley graph

Others with same cardinality

20 unique, 62 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	4095	⟨a, b \| baa=abb, abab=1⟩	Finite non-Abelian group with 20 elements	4 iso, 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	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.

3 total

Σ	#	Presentation	Mapping
11	18622	⟨a, b \| aba=b, aaaabba=1⟩	φ(a) = a, φ(b) = b
11	18628	⟨a, b \| aba=b, aaabbaa=1⟩	φ(a) = a, φ(b) = b
11	18668	⟨a, b \| aba=b, baaaaab=1⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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Next:	#18697 ⟨a, b \| aaa=a, aaabaa=b⟩