View source for 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
$$
\mathrm{core}_G (H) = \bigcap_g H^h \ ,\ \ \ H^g = gHg^{-1}
$$
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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,   "Group theory" , '''I''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.R. Scott,   "Group theory" , Dover, reprint  (1987)  (Original: Prentice-Hall, 1964)</TD></TR></table>

[[Category:Group theory and generalizations]]