Hi:

If A, B groups, M <= A, N <= B, then certainly M x N <= A x B (external direct product).

Now let A, B <= G, G = AB (internal direct product), M <= A, N <=B. Then

M x N <= A x B. (1)

In the elementary text books, at some point in the exposition the author says: "From now on we will no longer distinguish between external and internal direct products". Accordingly, (1) can be rewritten

MN <= AB, (2)

where both MN and AB are direct products. But to make a more formal inference, I'd have to use some isomorphism theorem.

I have one that gives that if MN is an internal direct product then MN isomorphic to M x N. As the elements of A commute with those of B, the elems. of M commute with those of N. This gives me, firstly, that MN <= G and, secondly, that M and N are normal in MN. Therefore, MN is direct. But now I do not need the isomorphism theorem because (2) follows straightforwardly.

The same would be true for the product of n subgroups of G. Is there a way to get (2) using only isomorphisms?