12. A normal subgroup H of a group G is said to be a direct factor (direct summand if

G is additive abelian) if there exists a (normal) subgroup K of G such that

G = HX K.

Prove that:

(a) If H is a direct factor of K and K is a direct factor of G, then H is normal

in G.

(b) If H is a direct factor of G, then every homomorphism H —>G may be

extended to an endomorphism G -> G. However, a monomorphism H -> G need

not be extendible to an automorphism G —> G.