$G_i$ s are normal subgroups , then $\bigl[G:\bigcap_{i=1}^n G_i \bigr]\Bigm|\prod_{i=1}^n[G:G_i]$?

Question

If $G_i$ $(i=1,\dots,n$) are normal subgroups of $G$, of finite index, then is it true that

$\displaystyle\bigl[G:\bigcap_{i=1}^n G_i \bigr]\Bigm|\prod_{i=1}^n[G:G_i]$?

Answer 1

Yes, define a homomorphism $$\phi: G \rightarrow G/G_1 \times G/G_2 \times \dots \times G/G_n$$ by $\phi(g)=(gG_1, gG_2, \dots, gG_n)$. Oberve that $\ker(\phi)=\bigcap_{i=1}^n G_i$, so by one of the isomorphism theorems, $G/\ker(\phi)$ is isomorphic to a subgroup, namely $\operatorname{im}(\phi)$, of the (finite) group $G/G_1 \times G/G_2 \times \dots \times G/G_n$. Now apply Lagrange's Theorem.