Normal p-complement
= Normal p-complement
{title2=$K\triangleleft G,\quad p\nmid|K|,\quad [G:K]=|P|$}
For a finite group with Sylow $p$-subgroup $P$, a normal $p$-complement is a normal subgroup of order $|G|/|P|$. It consists exactly of the elements of order prime to $p$: every such element has trivial image in the $p$-group quotient, and every element of the complement has prime-to-$p$ order. Thus a normal $p$-complement, if it exists, is unique and a <characteristic subgroup>.