To find a possible group G satisfying the given property, let's break down the problem into smaller parts.


The given property states that for any homomorphism f: G → Z221, the image of any element g in G under f is always 0. In other words, the homomorphism f maps every element of G to the identity element 0 in the additive group Z221.


To understand this property better, let's review some key properties of homomorphisms:

1. A homomorphism preserves the group operation. That is, for any elements a and b in G, f(a * b) = f(a) + f(b).


2. The identity element in the domain group G is mapped to the identity element in the codomain group Z221. That is, f(e) = 0, where e is the identity element of G.


3. The inverse of an element in G is mapped to the inverse of its image in Z221. That is, for any g in G, f(g⁻¹) = -f(g).

Based on the given property, we need to find a group G for which every homomorphism from G to Z221 maps every element to 0.

One such example is the trivial group, {e}, which consists of only the identity element. In this case, any homomorphism from the trivial group to Z221 will satisfy the given property because there is only one element to map, and it will always be mapped to 0.

Another example is the group of integers under addition, denoted by (Z, +). In this group, the identity element is 0, and every element has an additive inverse. Therefore, any homomorphism from (Z, +) to Z221 will also satisfy the given property because the inverse of any element in Z is also in Z.


In conclusion, one possible group G that satisfies the given property is the trivial group {e}, consisting of only the identity element. Another possible group is the group of integers under addition (Z, +). In both cases, any homomorphism from these groups to Z221 will map every element to 0.