Examples of properties not preserved under homomorphism An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the second.
However, homomorphisms only indicate that the two structures are "similar", so it's not quite as obvious that every property will be preserved. Yet all the properties I've ever seen are preserved under homomorphism: commutativity, cyclicality, solvability...
What are some examples of properties of algebraic structures not preserved under homomorphism? Feel free to use any algebraic structures you like, but I'm particularly interested in your garden variety structures: group and rings, say.

One way to get some properties not preserved under general homomorphisms is to take quantitative versions of some qualitative properties that are so preserved.  For instance, in the case of groups, while solubility is preserved, the derived length is not.  Likewise, nilpotence is preserved but the class of nilpotence is not.  The exponent of a group is not preserved (but being of finite exponent is).
A few  others that come to mind (for groups): residual finiteness (and lots of other residual properties), being centre-less, being free and being torsion-free.
