# When Does a Map Induce an Isomorphism or Homomorphism?

• WWGD
In summary, the conversation is about the use of "induced homomorphs" and "induced isomorphisms" in homological algebra and the lack of a formal result explaining under what conditions a map induces an isomorphism or homomorphism. It is mentioned that the induced maps are typically defined in some quotient space of the domain and that the map in question is usually continuous. The speaker is looking for a result that would allow them to determine if a map induces an isomorphism or homomorphism. It is also mentioned that in some cases, maps are induced when considering categories associated with spaces, such as in homology and homotopy.
WWGD
Gold Member
Hi, everyone:
I keep seeing, mostly in homological algebra, the use of "induced
homomorphs" or "induced isomorphisms". I get the idea of what is
going on, but I have not been able to find a formal result that
rigorously explains this, i.e, under what conditions does a map
induce an isomorphism or a homomorphism?. The only patterns
there seems to be in all these induced maps is that they are all
defined in some quotient space of the domain, i.e, if
we have f:X->Y , then the induced maps f* are , or seem to be,
defined in X/~ , for some equiv. relation ~ (e.g, in homotopy and
homology). Also, maybe obviously, f is a continuous map.

Basically: I would like to know a result that would allow me to
give a yes/no answer to the question : does f:X->Y induce an
isomorphism/homomorphism of some sort?

Thanks.

Just in case someone else is interested, I think I have at least a partial answer,
i.e, 2 cases in which maps are induced:

In some cases, maps are induced when we consider categories associated with
spaces, i.e, if we are given a map f (continuous) , f:X->Y , we consider (f_x,O_x)
and (f_y,O_y) as categories , i.e, f_x,f_y are morphisms, and O_x,O_y are
objects, respectfully (of course, we can only do this in some cases only).

As an example, we would consider homology and homotopy as functors.
Anyway, that is a sketch.

## 1. What are induced homomorphs and isomorphs?

Induced homomorphs and isomorphs are mathematical concepts used in group theory to describe relationships between two groups. An induced homomorphism is a mapping between two groups that preserves the group structure, while an isomorphism is a type of homomorphism that is bijective, meaning it has a one-to-one correspondence between elements of the two groups.

## 2. How do induced homomorphs and isomorphs differ?

Induced homomorphs and isomorphs differ in their degree of flexibility. While an induced homomorphism must preserve the group structure, an isomorphism can also preserve the specific elements and operations within the groups. This means that an isomorphism is a more specific type of induced homomorphism.

## 3. How are induced homomorphs and isomorphs used in scientific research?

Induced homomorphs and isomorphs are used in scientific research to study the structure and behavior of groups, such as symmetries in molecules or patterns in mathematical equations. They can also be used to classify and compare different groups, which can provide insight into the underlying properties of these structures.

## 4. What is an example of an induced homomorphism?

An example of an induced homomorphism is the mapping between the integers and the even integers, where the group structure of addition is preserved. This means that adding two even integers will always result in an even integer, and adding an even and odd integer will result in an odd integer.

## 5. How do induced homomorphs and isomorphs relate to other mathematical concepts?

Induced homomorphs and isomorphs are closely related to other mathematical concepts such as group actions, subgroups, and quotient groups. In fact, they can be used to define these concepts and provide a deeper understanding of their properties and relationships.

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