# Number of group homomorphisms from Ｚ

• hmw
In summary, the number of group homomorphisms from Zn to Zm is equal to the greatest common divisor (gcd) of n and m. This is because all homomorphisms can be expressed as f([x])=[kx], where k is an integer that satisfies the condition m|kn. This means that the number of possible values for k is equal to the gcd of n and m, and thus, the number of group homomorphisms is also equal to the gcd of n and m.
hmw

## Homework Statement

Show that the number of group homomorphisms from Zn to Zm is equal to gcd(n,m).

my attempt:

any hom from Zn to Zm must be f([x])=[kx] where k is a common factor of n and m. I can only get this far... any help is appreciated.

Last edited:
As you've said, all homomorphism are of the form f([x])=[kx], with k an integer. Of course, these maps are only well-defined for certain choices of k. However, it turns out the criteria is not that k be a common factor of n and m.

Specifically, we must have:

x=y (mod n) => kx=ky (mod m)

which its pretty easy to see is equivalent to:

n|z => m|kz

This will be satisfied iff m divides kn. One obvious choice is k=0, which gives the trivial map sending all [x] to [0]. This homomorphism always exists. Another choice is k=m, but this is equivalent to the trivial map since [mx]=[0] in Z_m. In general, we only need to find all the solutions k with 0<=k<m, since solutions differing by a multiple of m are easily seen to give the same map.

Nontrivial values for k will only exist if gcd(n,m)>1. For example, if n=4 and m=6, we can take k=3. Try a few more examples, and hopefully you'll see the pattern that emerges.

## 1. How do you calculate the number of group homomorphisms from ℤ?

The number of group homomorphisms from ℤ can be calculated using a simple formula: nm, where n is the number of elements in the domain group (in this case, ℤ) and m is the number of elements in the codomain group.

## 2. What is the significance of the number of group homomorphisms from ℤ?

The number of group homomorphisms from ℤ is important because it represents the number of possible ways to map elements from the domain group to the codomain group while preserving the group structure. This can provide valuable insights into the structure and properties of the two groups.

## 3. Can the number of group homomorphisms from ℤ be negative?

No, the number of group homomorphisms from ℤ cannot be negative. The formula used to calculate this number involves an exponent, which means the result will always be a positive integer or zero.

## 4. How does the number of group homomorphisms from ℤ change when the size of the domain or codomain group is increased?

The number of group homomorphisms from ℤ increases exponentially when the size of the domain or codomain group is increased. This is because the formula used to calculate the number involves an exponent, meaning that even a small increase in the size of the groups can result in a significant increase in the number of homomorphisms.

## 5. Is the number of group homomorphisms from ℤ always finite?

No, the number of group homomorphisms from ℤ is not always finite. It depends on the size of the domain and codomain groups. If both groups are finite, then the number of homomorphisms will also be finite. However, if either group is infinite, then the number of homomorphisms will also be infinite.

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