TITLE: The kernel of a group action defined by left multiplication on left cosets of $H\leq G$
QUESTION [9 upvotes]: Given a subgroup $H\leq G$, for $G$ finite.  Let $G$ act on the left cosets of $H$ by left multiplication.  What is the kernel of this group action?
Since the left cosets are themselves constructed by left multiplication, wouldn't the kernel of this map just be equal to $H$?  Am I missing something?

REPLY [3 votes]: The kernel is what is called the core of $\,H\,$ = the largest normal subgroup of $\,G\,$ contained in the subgroup $\,H\,$ , and it equals
$$cor(H):=\bigcap_{g\in G}H^g=\bigcap_{g\in G}g^{-1}Hg$$