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I was wondering if the following statement is true:

Let $G$ be a group with normal subgroups $H_1,H_2,...H_n$. Suppose $H_iH_j=G$ for all $ineq j$. Then $G/H_1cap H_2...cap H_ncong G/H_1 times...times G/H_n.$

There is a similar question here, which is the case when $n=2$. There is also a similar question here, but the conditions involve the index of a subgroup. I want to get rid of such condition so that the conclusion is applicable to some other situations.

But when I tried to proceed proof by induction to get the conclusion with arbitrary $n$, things became not so approachable.

If you think this is false, please give a counter-example. If you think this is true, please share your ideas of the proof. Thank you!

This is false. For instance, let $G=(mathbbZ/2mathbbZ)^2$ and let $H_1,H_2,H_3subset G$ be the three subgroups of order $2$. Then $H_iH_j=G$ for all $ineq j$ but $G/(H_1cap H_2cap H_3)cong G$ has $4$ elements while $G/H_1times G/H_2times G/H_3cong(mathbbZ/2mathbbZ)^3$ has $8$ elements.

Here's a counterexample . . .

Let $G=Z_n$, where $n=3cdot 5cdot 11$, regarded as a multiplicative group, and let $A,B,C$ be the subgroups of $G$ given by
$$A=langlex^3rangle,;;;B=langlex^5rangle,;;;C=langlex^11rangle$$
where $x$ is a cyclic generator of $G$.

Then we have
beginalign*
AB&=langlex^8rangle=G\[4pt]
BC&=langlex^16rangle=G\[4pt]
CA&=langlex^14rangle=G\[4pt]
endalign*
and then, since $Acap Bcap C=1$, we get
$$LargeG/(Acap Bcap C)Large=|G|=n$$
but
$$
Large(G/A)times(G/B)times(G/C)Large
=
fracn3cdot fracn5cdotfracn11
=fracn^3n=n^2 > n
$$