I don't quite understand this sentence: "Every external direct product is naturally realized as an internal direct product."
Does "realize" mean that $H\times K$ is equal to $H+K$ (where $H,K$ are groups and + between groups means internal product) ? Or does it mean $H\times K$ is isomorphic to $H+K$ ?
I tried to prove these two are isomorphic, with the isomorphism $\phi: H\times K \rightarrow H+K$ such that $\phi: X \mapsto \pi_1 (X) \cdot \pi_2 (X)$ where $\pi_1 , \pi_2 $ are the projections from $H\times K$ to $H$ and $K$ respectively.
However, I don't think that what I wrote in my $2^{nd}$ paragraph (namely $H+K$) is allowed. We can only write $H+K$ if they are subgroups of a wider group $G$ and if three conditions are realized:
- $G = HK = \{hk | h\in H, k\in K \}$
- $H\cap K = \{ 1_{G} \}$
- $hk = kh$ for every $h\in H, k\in K $
So is my isomorphism $\phi$ enough or should I go and prove all three points to prove $H\times K$ is an internal direct product (made of $H$ and $K$) ?
Thanks in advance
I think "realized" here means you can somehow regard $H \times K$ as an internal direct product of some group. The point is where this internal direct product is. Let's say you are working over ambient group $G = H \times K$. This group $G$ needs not contain $H$ or $K$, but it contains a copy of $H$, namely $H' = H \times {e_K}$. Similarly $K'=e_H \times K$ is a copy of $K$. Then it can be checked that $H', K'$ satisfied three conditions you listed above, thus $H \times K$ can be thought as internal direct product of $H'K'$ in $G$.