Abstract:
We classify up to isomorphism the normal subgroups of free profinite groups and also of their analogues, the so-called free pro-$\Delta$-groups, which include free prosoluble groups and free pro-$\pi$-groups (where $\pi$ is a set of primes). We prove that if $N$ is a normal subgroup of a free ðãî-$\Delta$-group, then any proper normal subgroup of $N$ of finite index is a free ðrî-$\Delta$-group. We find a set of conditions that are comparatively easy to check, which guarantee the freeness of a normal subgroup of a free pro-$\Delta$-group. We discuss the question of when a normal subgroup of a free ðrî-$\Delta$-group is determined by the set of its finite homomorphic images.
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