Question

Show that an abelian group is the direct product of its p-sylow subgroups for primes p dividing |g|.

Answer 1

This direct product de- composition is unique, up to a reordering of the factors. be the order of the Abelian group G, with pi's distinct primes. BySylow's theorem it follows that G has exacly one Sylow p−subgroup for each of the k distinct primes pi. Consequently G is the direct product of its Sylow p−subgroups.