Core of a subgroup

Let $H$ be a subgroup of $G$. The core of $H$ is the maximal subgroup of $H$ that is a normal subgroup of $G$. It follows that $$ \mathrm{core}_G (H) = \bigcap_g H^g \ ,\ \ \ H^g = gHg^{-1} $$ If the index $[G:H] = n < \infty$, then $[G:\mathrm{core}_G (H)]$ divides $n!$.

Let $g(xH) = (gx)H$ and define the permutation representation of $G$ on the set of right cosets of $H$ in $G$ (cf Coset in a group). Then its kernel is the core of $H$ in $G$.

References

[a1]	M. Suzuki, "Group theory" , I , Springer (1982)
[a2]	W.R. Scott, "Group theory" , Dover, reprint (1987) (Original: Prentice-Hall, 1964)