  We establish some general principles and find some counter-examples
concerning the Pontryagin reflexivity of precompact groups and P-groups. We
prove in particular that: (1) A precompact Abelian group G of bounded order is
reflexive iff the dual group $\hat{G}$ has no infinite compact subsets and
every compact subset of G is contained in a compact subgroup of G. (2) Any
extension of a reflexive P-group by another reflexive P-group is again
reflexive. We show on the other hand that an extension of a compact group by a
reflexive $\omega$-bounded group (even dual to a reflexive P-group) can fail to
be reflexive. We also show that the P-modification of a reflexive
$\sigma$-compact group can be nonreflexive (even if the P-modification of a
locally compact Abelian group is always reflexive).
