# Inverse image of a homomorphism

• tomkoolen
In summary, if f: G -> H is a homomorphism of groups with a finite kernel of n elements, the inverse image of any element in H is either empty or has exactly n elements. This can be shown by considering the inverse image of the unit element, eH, and noting that if f(x) = f(y), then f(xy^-1) = e, which implies that the inverse image of any element in H must contain n elements.
tomkoolen
Member warned about posting without the HW template
The question: Let f: G -> H be a homomorphism of groups with ker(f) finite, the number of elements being n. Show that the inverse image is either empty or has exactly n elements.

My work so far:
Let h be eH (identity on H). Then the inverse image is ker(f) so has n elements, which makes it empty when n = 0.
I know how to do this for eH but how for all other y in H?

What makes you think any homomorphism is invertible?

EDIT: Scratch that, I might have mis-understood.
What is an inverse image, exactly? Do you mean the subset of H that is mapped from G?This is the kernel
$Ker f := \{g\in G | f(g) = e\in H\}$
assuming I understand the problem correctly
we also know that $|Ker f| = n$

The inverse image of just the unit element: $f^{-1} (\{e\}) = \{e\in G, g_2,\ldots g_n\}\subset G$

Last edited:
tomkoolen said:
The question: Let f: G -> H be a homomorphism of groups with ker(f) finite, the number of elements being n. Show that the inverse image is either empty or has exactly n elements.

My work so far:
Let h be eH (identity on H). Then the inverse image is ker(f) so has n elements, which makes it empty when n = 0.
I know how to do this for eH but how for all other y in H?
I assume you meant that the inverse image of an element of H is either empty or has exactly n elements.

If ##f(x)=f(y)##, what is ##f(xy^{-1})##?
What does the answer tell you about the set ##D=f^{-1}(h)##, where ##h \in Im(f)##?

## 1. What is an inverse image of a homomorphism?

The inverse image of a homomorphism refers to the set of all elements in the domain of the homomorphism that map to a specific element in the codomain.

## 2. How is the inverse image of a homomorphism different from the image?

The image of a homomorphism refers to the set of all elements in the codomain that are mapped to by at least one element in the domain. The inverse image, on the other hand, refers to the set of all elements in the domain that are mapped to a specific element in the codomain.

## 3. What is the significance of the inverse image of a homomorphism?

The inverse image of a homomorphism is significant because it helps to identify which elements in the domain are responsible for mapping to a specific element in the codomain. This is useful in understanding the structure and behavior of the homomorphism.

## 4. How is the inverse image of a homomorphism related to the preimage?

The inverse image of a homomorphism is similar to the preimage, which refers to the set of all elements in the codomain that map to a specific subset of the domain. However, the inverse image is more specific as it refers to the set of all elements in the domain that map to a specific element in the codomain.

## 5. Can the inverse image of a homomorphism be empty?

Yes, the inverse image of a homomorphism can be empty if there are no elements in the domain that map to a specific element in the codomain. This can happen if the homomorphism is not surjective, meaning that not all elements in the codomain are mapped to by at least one element in the domain.

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