Examples of properties not preserved under homomorphism

A very simple example is cardinality.

An image of an algebraic object is equivalently a quotient in the most elementary cases. Taking a quotient is an identification process, so a general class of properties not preserved under images are those relating to uniqueness of solutions of equations.

For instance, in any free abelian group a linear equation with a solution has only one solution-but in abelian groups with torsion there may be many. Similarly, rings of polynomials over a field admit factorization theorems to the effect that a polynomial of degree $n$ has no more than $n$ roots, whereas there are nonzero polynomials over finite rings that annihilate the entire ring.