Let $H$, $K$ be subgroups of a given group $G$. Can one show that $(G:(H\cap K))$ is less or equal to $(G:H)(G:K)$, where $(G:H)$ stands for the index of group G with respect to $H$?
[Math] Index of a group
abstract-algebragroup-theory
abstract-algebragroup-theory
Let $H$, $K$ be subgroups of a given group $G$. Can one show that $(G:(H\cap K))$ is less or equal to $(G:H)(G:K)$, where $(G:H)$ stands for the index of group G with respect to $H$?
Best Answer
You could show that $a(H \cap K) = aH \cap aK\ $ for every $a\in G$. We can choose $aH \cap aK$ in $[G:H][G:K]$ ways. Some of the combinations might be same, but there can be no more than $[G:H][G:K]$ cosets of $H \cap K$.