If G and H are non-abelian simple groups and N is a normal subgroup of G×H, then N is equal (not just isomorphic) to 1×1, 1×H, G×1, or G×H. If G and H are simple groups but not isomorphic, then the same is true. If G and H are isomorphic abelian simple groups, then G×H is a two-dimensional vector space over some Z/pZ, and so it has 1 + (p+1) + 1 normal subgroups, 1 of dimension 0, p+1 of dimension 1, and 1 of dimension 2. Only in the last case can it happen that N is not precisely equal to (and also not isomorphic to) π₁(N)×π₂(N), but of course this happens for p-1 of the 1-dimensional subspaces, the ones that have non-trivial projection onto both the x and y axes.

If $G$ and $H$ are non-abelian simple groups and $N$ is a normal subgroup of $G×H$, then $N$ is equal (not just isomorphic) to $1×1$, $1×H$, $G×1$, or $G×H$. If $G$ and $H$ are simple groups but not isomorphic, then the same is true. If $G$ and $H$ are isomorphic abelian simple groups, then $G×H$ is a two-dimensional vector space over some $\Bbb Z/p\Bbb Z$, and so it has $1 + (p+1) + 1$ normal subgroups, $1$ of dimension $0$, $p+1$ of dimension $1$, and $1$ of dimension $2$. Only in the last case can it happen that $N$ is not precisely equal to (and also not isomorphic to) $π_1(N)×π_2(N)$, but of course this happens for $p-1$ of the $1$-dimensional subspaces, the ones that have non-trivial projection onto both the $x$ and $y$ axes.

If G and H are non-abelian simple groups and N is a normal subgroup of G×H, then N is equal (not just isomorphic) to 1×1, 1×H, G×1, or G×H. If G and H are abelian simple groups but not isomorphic, then the same is true. If G and H are isomorphic abelian simple groups, then G×H is a two-dimensional vector space over some Z/pZ, and so it has 1 + (p+1) + 1 normal subgroups, 1 of dimension 0, p+1 of dimension 1, and 1 of dimension 2. Only in the last case can it happen that N is not precisely equal to (and also not isomorphic to) π₁(N)×π₂(N), but of course this happens for p-1 of the 1-dimensional subspaces, the ones that have non-trivial projection onto both the x and y axes.

If G and H are non-abelian simple groups and N is a normal subgroup of G×H, then N is equal (not just isomorphic) to 1×1, 1×H, G×1, or G×H. If G and H are simple groups but not isomorphic, then the same is true. If G and H are isomorphic abelian simple groups, then G×H is a two-dimensional vector space over some Z/pZ, and so it has 1 + (p+1) + 1 normal subgroups, 1 of dimension 0, p+1 of dimension 1, and 1 of dimension 2. Only in the last case can it happen that N is not precisely equal to (and also not isomorphic to) π₁(N)×π₂(N), but of course this happens for p-1 of the 1-dimensional subspaces, the ones that have non-trivial projection onto both the x and y axes.