Homotopy classes of self-maps and induced homomorphisms of homotopy groups

For a based space X, we consider the group E-#n(X) of all self homotopy classes alpha of X such that alpha(#) = id : pi(i) (X) -> pi(i)(X), for all i <= n, where n <= infinity, and the group E-Omega (X) of all alpha such that Omega alpha = id. Analogously, we study the semigroups F-#n (X) and F-Omega(X) defined by replacing 'id' by '0' above. There is a chain of containments of the E-groups and the F-semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on X which ensure that the E-groups and the F-semigroups are equal. When X is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for the E-groups and F-groups of certain low dimensional Lie groups.