An arithmetic property of profinite groups

We intend to generalize a crucial lemma of 4] to prove a somewhat surprising arithmetic property of profinite groups; namely, that a profinite group G has nontrivial p-Sylow-subgroups for only a finite number of primes if and only if this is true for its procyclic subgroups. This will yield as a corollary that every profinite torsion group has finite exponent if and only if this is true for its Sylow-sub-groups, a result also contained in 4].