Example of Group that is not Internal direct product under weaker condition

Question

I know definition of internal direct which is stronger that is

$G$ is internal direct product of normal subgroup $N_i$ where $i$ is indexing upto $n$. Then

1. $G=N_1N_2N_3N_4N_5......N_n$
2. $N_i \cap (N_1....N_{i-1}N_{i+1}......N_n)=(e)$

It is clear that internal direct product is not possible in case of weaker condition of $N_i \cap N_j=(e)$. But I couldn't able to find counterexample to justify above.

Any help in this regard will be appreciated.

Answer 1

Sure, consider the Klein four group. It has three two-element subgroups, and each pairwise intersection is trivial, but it is not a direct product of all three, or it would have order 8 rather than order 4.