ABSTRACT
Suppose that (2.1) is an indecomposable decomposition of G and suppose that we can also write
G ∼= H ⊕K ∼= H ⊕K ′. (2.2)
We will pursue several broad questions on the nature of direct sums. For instance, is the direct sum (2.1) unique in some sense? What can be said about K and K ′ in (2.2)? Is H a direct sum of copies of the Gi? Are the group structures of K and K ′ similar in some sense? Are there examples of badly behaved direct sum decompositions of G? Is there a be
n > 0. If G is a free group then pi is split. That is, is there a map φ : G→ G(n) such that piφ = 1G, or equivalently, such that kerpi a direct summand of G(n). Under what conditions on G will pi be split?
