An abelian group G is said to be a p-local group if G is a module over the ring \({\mathbb{Z}}_ p=\{m/n|\) \(m,n\in {\mathbb{Z}}\), \((n,p)=1\}\). An exact sequence \(0\to A\to B\to C\to 0\) is called balanced exact if the sequence \(0\to p^{\alpha}A\to p^{\alpha}B\to p^{\alpha}C\to 0\) is also exact for each ordinal \(\alpha\), and a p-local group G is called balanced projective if G satisfies the projective property with respect to all balanced exact sequences. If \(x\in G\), then the p-height sequence \({\bar \alpha}=\{\alpha_ n\}_{n<\omega}\) of x is the sequence, in which \(\alpha_ n=| p^ nx|\) (by \(| p^ nx|\) we denote the p-height of the element \(p^ nx)\). For every strictly increasing sequence of ordinals and symbols \(\infty\) \({\bar \alpha}=\{\alpha_ n\}_{n<\omega}\) (if \(\alpha_ k=\infty\), then \(\alpha_{k+1}=\alpha_{k+2}=...=\infty)\), there is a fully invariant submodule (\({\bar \alpha}\))\(=\{x\in G|\) \(| p^ nx| \geq \alpha_ n\) for every \(n<\omega \}\). The sequence \({\bar \alpha}\) is said to be a p-height sequence. It is an unpublished result attributed to E. Walker and L. Fuchs that every fully invariant subgroup of a totally projective p-group G has the form G(\({\bar \alpha}\)). A complete description of the structure of fully invariant subgroups of torsion free abelian groups, which are modules over the ring of p-adic integers was obtained by S. Ya. Grinshpon [Abelian groups and modules, Tomsk 1981, 56-92 (1981; Zbl 0527.20043)].
In the present paper the author gives a complete description of fully invariant submodules of p-local balanced projective groups. The main results of the paper are as follows. Theorem 1. If \({\bar \alpha}\) is a p-height sequence and G is balanced projective, then G(\({\bar \alpha}\)) is also balanced projective. Theorem 2. If H is a fully invariant submodule of the reduced p-local balanced projective group G, then H is an SKT module. SKT modules are defined as p-local groups which are isomorphic to a direct sum of S-groups (i.e. the torsion submodule of a p-local balanced projective group) and a balanced projective group.