  We give algebraic characterizations for expansiveness of algebraic actions of
countable groups. The notion of p-expansiveness is introduced for algebraic
actions, and we show that for countable amenable groups, a finitely presented
algebraic action is 1-expansive exactly when it has finite entropy. We also
study the local entropy theory for actions of countable amenable groups on
compact groups by automorphisms, and show that the IE group determines the
Pinsker factor for such actions. For an expansive algebraic action of a
polycyclic-by-finite group on X, it is shown that the entropy of the action is
equal to the entropy of the induced action on the Pontryagin dual of the
homoclinic group, the homoclinic group is a dense subgroup of the IE group, the
homoclinic group is nontrivial exactly when the action has positive entropy,
and the homoclinic group is dense in X exactly when the action has completely
positive entropy.
