I have a theorem states that:

Let$\displaystyle G=H\times K$, and let$\displaystyle H_{1}\triangleleft H$and$\displaystyle K_{1}\triangleleft K$.Then$\displaystyle H_{1}\times K_{1}\triangleleft G$and$\displaystyle G/(H_{1}\times K_{1})\cong(H/H_{1})\times(K/K_{1})$.

Then I have this corollary:

If$\displaystyle G=H\times K$,then$\displaystyle G/(H\times\{{1}\})\cong K$.

Please check for me if my proof for the corollary is correct.

Proof:

We consider$\displaystyle H_{1}=H$,i.e. H has no proper normal subgroup.$\displaystyle K_{1}=\{1\}$.

Also,

By last theorem, $\displaystyle G/(H_{1}\times K_{1})\cong (H/H_{1})\times(K/K_{1})$.

We have$\displaystyle G/(H\times\{1\})\cong (H/H)\times(K/\{1\})$.

Since H has no proper normal subgroup, so quotient group of H is not defined. Also,$\displaystyle (K/\{1\})=K$.

Hence,$\displaystyle G/(H\times\{1\})\cong K$