Is there a subgroup $H$ of $G$ of index $p$, where $p$ is the smallest prime dividing $|G|<\infty$? [closed]
To be clear: I'm not asking whether $H$ is normal in $G$ or not. It is well-known that if such $H$ exists, then $H$ is normal in $G$, which is out of my interest right now.
But can we say that such $H$ always exists? (for any finite group $G$?)
$A_{4}$ does not have a subgroup of order $6$.
To elaborate , $A_{4}=2^{2}\cdot 3$
Since it does not have a subgroup of order $6$ . We cannot say that it has a subgroup of index $2$ which is the smallest prime dividing the order of $A_{4}$.