estimation of index of intersection subgroup

Theorem.  If $H_1,H_2,\mathrm{\dots },H_n$ are subgroups of $G$, then

$$[G:\bigcap _{i=1}^n{H}_i]\leqq \prod _{i=1}^n[G:H_i].$$

Proof.  We prove here only the case  $n=2$;  the general case may be handled by the induction.

Let  $H_1\cap H_2:=K$.  Let $R$ be the set of the right cosets of $K$ and $R_i$ the set of the right cosets of$H_i$  ($i=1,\mathrm{\hspace{0.17em}2}$).  Define the relation $\varrho$ from $R$ to $R_1\times R_2$ as

$$\varrho :=\{(Kx,(H_{1}x,H_{2}x))\mathrm{⋮}x\in G\}.$$

We then have the equivalent (http://planetmath.org/Equivalent3) conditions

$$Kx=Ky,$$

$$x{y}^{-1}\in K,$$

$$x{y}^{-1}\in H_1\mathit{\hspace{1em}}\wedge \mathit{\hspace{1em}}x{y}^{-1}\in H_2,$$

$$H_{1}x=H_{1}y\mathit{\hspace{1em}}\wedge \mathit{\hspace{1em}}{H}_{2}x=H_{2}y,$$

$$(H_{1}x,H_{2}x)=(H_{1}y,H_{2}y),$$

whence $\varrho$ is a mapping and injective,  $\varrho :R\to R_1\times R_2$.  i.e. it is a bijection from $R$ onto the subset  $\{\varrho (x)\mathrm{⋮}x\in R\}$  of $R_1\times R_2$.  Therefore,

$$\mathrm{card}(R)\leqq \mathrm{card}(R_1\times R_2)=\mathrm{card}(R_1)\cdot \mathrm{card}(R_2).$$

As a consequence one obtains the

Theorem (Poincaré).  The index of the intersection of finitely many subgroups with finite indices (http://planetmath.org/Coset) is finite.