Vtyjkyuk

Let $G$ and $H$ be two groups and let $Goplus H$ be the direct product of them. What is(are) the necessary and sufficient condition(s) for which each subgroup of the group $Goplus H$ will be of the form $Aoplus B$, where $A$ and $B$ are subgroups of $G$ and $H$ respectively?

One necessary condition is of course that $H$ is not a subgroup of $G$ and $G$ is not a subgroup of $H$ because otherwise then we will have the diagonal subgroup of $Hoplus H$ or $Goplus G$, respectively; which is not of the mentioned form. But is this condition sufficient?

Let $phi_G:Gtimes Hto G$ and $phi_H:Gtimes Hto H$ be the two projection maps.

Then any $Kle Gtimes H$ is a subdirect product of $phi_G(K)timesphi_H(K)$. It is the direct product required if and only if $Hcap K=phi_H(K)$ and $Gcap K=phi_G(K)$.

Goursat's Lemma tells us that $phi_H(K)/(Hcap K)congphi_G(K)/(Gcap K)$. So if $K$ is not the direct product then $G,H$ have some non-trivial isomorphic subquotients.

Conversely suppose $Ntrianglelefteq Kle G$ and $Mtrianglelefteq Lle H$ with $K/N$ non-trivial and an isomorphism $phi:K/Nto L/M$. Then the set $cup_gin K(g,phi(g)M)$ where $phi(g)M=lin L$ is a subgroup of $Gtimes H$ and not a direct product as described.

That is all subgroups of $Gtimes H$ are of the form $Atimes B$ with $Ale G$, $Ble H$ if and only if $G$ and $H$ have no isomorphic subquotient.