I'm working on the following question:
If $H$ and $K$ are normal subgroups of $G$, and $H\cap K = \{id\}$, then show that $G$ is isomorphic to a subgroup of the direct product group $\frac{G}{H} \times \frac{G}{K}$
I'm really not sure how to proceed with this. I'm familiar with the Isomorphism Theorems, and this question comes from the same section of my textbook, so perhaps they're related. However I don't see how to relate this to a direct product group, much less a direct product.
I also know that for two subgroups $H,K$, if elements of $H$ and $K$ commute with each other and $G=HK$, then $G$ is isomorphic to $H \times K$.
I'm just not sure how to put things together here. A hint or two would be hugely appreciated.
Best Answer
Sketch: The way that these problems always go is that you need to define a map. There is a natural map $$ G\rightarrow(G/H)\times (G/K) $$ given by the product of the quotient maps. Now, you only need to show that the map is injective.
So, suppose that $g_1$ and $g_2$ go to the same place under the product of the quotients. Therefore, $g_1H=g_2H$ and $g_1K=g_2K$ (assuming left cosets). Now, can you show that $g_1^{-1}g_2\in H\cap K$, so $g_1=g_2$?