You basically just write it down carefully. It is a matter of notation rather than a matter of mathematics. It is constructed to fulfill the universal property.Heidi said:
Why don't you try what I have said? Write down all morphisms that you have, name the group and the index set, and note what must be shown. I see no sense in rewriting the definition. So in order to communicate, we need a common language, not just M*N. Have you read the nLab link?Heidi said:
Ok, this simply means ##H=\{1\}.## The elements of ##G=G_1\ast G_2## are words over the alphabet of elements from ##G_1## and ##G_2.## Say we have ##abcxyacbza\in G## with ##a,b,c\in G_1## and ##x,y,z\in G_2.## Then we have a homomorphism ##f_1\, : \,G_1\longrightarrow Q## that maps ##a,b,c,## say onto ##\overline{a},\overline{b},\overline{c}## and the same for ##f_2\, : \,G_2\longrightarrow Q## that maps ##x,y,z## onto ##x',y',z'.## The dashed group homomorphism ##f## maps then ##abcxyacbza## onto ##f_1(a)f_1(b)f_1(c)f_2(x)f_2(y)f_1(a)f_1(c)f_1(b)f_2(z)f_1(a)= \overline{a}\overline{b}\overline{c}x'y'\overline{a}\overline{c}\overline{b}z'\overline{a}.##Heidi said:
The universal property is a concept in mathematics that describes a unique mapping between two mathematical objects. It states that a particular construction or object satisfies a certain property that is universal among all other objects with the same property.
In order to verify the universal property, one must show that the construction or object in question satisfies the defining property of the universal property. This can be done by proving that the construction or object has a unique mapping to all other objects with the same property.
Verifying the universal property is important because it allows us to prove the uniqueness of a certain construction or object. It also helps us understand the relationship between different mathematical objects and how they are connected through a universal property.
If the construction does not verify the universal property, it means that it does not satisfy the defining property of the universal property. This could mean that the construction is not unique or that it does not have a unique mapping to all other objects with the same property.
Yes, the universal property can be applied to all mathematical objects as long as they have a defining property that can be satisfied by a unique construction or object. It is a general concept that can be used in various areas of mathematics to prove uniqueness and establish connections between different objects.