A (sub)group in which every element has order a power of a fixed prime p is called a p-(sub)group. Let G be an abelian torsion group.(a) G(p) is the unique maximum p-subgroup of G (that is, every p subgroup of G is contained in G(p)).(b) where the sum is over all primes p such that G(p) ≠ 0.

(c) If H is another abelian torsion group, then G ≅ H if and only if G(p) ≅ H(p) for all primes p.

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G = G(p),