p primary group and the correspondence theorem

Hi, this is a question from " A first course in abstract algebra" by J. Rotman

define d(G) = dim(G/pG) (not very relevant here)

chapter 5, lemma 5.8 (P392),

Let G be a finite p primary abelian group.

If S<=G, then d(G/S) <= d(G)

The first line of the proof read like,

By the correspondence theorem, p(G/S) = (pG +S)/S,

How is this equation derived? As the correspondence theorem mainly states isomorphism, I cannot see where there is equation involved? It would be greatly appreciated if anyone could help on this. Many thanks!