Understanding the proof of " The external direct product is isomorphic to the internal direct product"

Question

Let $G$ be a group with $2$ subgroups $H$ and $K$ s.t

1. $H.K = G$
2. $H,K \triangleleft G$
3. $H \cap K = \{1\}$

Then $G \cong H \times K$ where $H \times K$ is the external direct product

My professor proved it in 3 parts:

• All elements of can be written uniquely as a product of an element of $H$ and an element of $K$.
• Elements of $H$ and elements of $K$ commute.
• Construction of an isomorphism between $G$ and $H\times K$

What I don't understand:

Why doesn't the construction of an isomorphism suffice itself?

Answer 1

How did your professor carry out the construction of the third bullet point, and prove the construction is what it's claimed to be?

Presumably, your professor made use of the results from the first two bullet points.

Or, at the very least, the results of the first two bullet points serve to motivate the construction.

Answer 2

Constructing an isomorphism between $G$ and $H \times K$ would certainly be sufficient. In fact, that is the definition of isomorphism between two groups.

On the other hand, however, in order to define the isomorphism you may (in fact you do) need to use what is proven by the first two bullets.

Your professor probably proved those two things first and then built the isomorphism taking them into consideration.