estimation of index of intersection subgroup

Theorem. 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
Canonical name	EstimationOfIndexOfIntersectionSubgroup
Date of creation	2013-03-22 18:56:46
Last modified on	2013-03-22 18:56:46
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	5
Author	pahio (2872)
Entry type	Theorem
Classification	msc 20D99
Synonym	index of intersection subgroup
Related topic	LogicalAnd
Related topic	Cardinality