# estimation of index of intersection subgroup

If $H_{1},\,H_{2},\,\ldots,\,H_{n}$ are subgroups of $G$, then

 $\left[G:\bigcap_{i=1}^{n}H_{i}\right]\leqq\prod_{i=1}^{n}[G:H_{i}].$

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 $H_{i}$  ($i=1,\,2$).  Define the relation $\varrho$ from $R$ to $R_{1}\!\times\!R_{2}$ as

 $\varrho\;:=\;\{\left(Kx,\,(H_{1}x,\,H_{2}x)\right)\vdots\;\;x\in G\}.$

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

 $Kx\;=\;Ky,$
 $xy^{-1}\in K,$
 $xy^{-1}\in H_{1}\quad\land\quad xy^{-1}\in H_{2},$
 $H_{1}x\;=\;H_{1}y\quad\land\quad H_{2}x\;=\;H_{2}y,$
 $(H_{1}x,\,H_{2}x)\;=\;(H_{1}y,\,H_{2}y),$

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  $\{\varrho(x)\vdots\;\;x\in R\}$  of $R_{1}\!\times\!R_{2}$.  Therefore,

 $\operatorname{card}(R)\;\leqq\;\operatorname{card}(R_{1}\!\times\!R_{2})\;=\;% \operatorname{card}(R_{1})\cdot\operatorname{card}(R_{2}).$

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.

Title estimation of index of intersection subgroup EstimationOfIndexOfIntersectionSubgroup 2013-03-22 18:56:46 2013-03-22 18:56:46 pahio (2872) pahio (2872) 5 pahio (2872) Theorem msc 20D99 index of intersection subgroup LogicalAnd Cardinality