about a maximal normal subgroup of a $p$ group. i'm studying bhattacharya's basic algebra. it introduces the concept of the group action in chapter 4 and proves the class equation. and derives simple properties of $p$ group using the equation. the author proves the thm stating :
"Let $G$ be a finite group of order $p^{n}$, where $p$ is a prime $n$ is a natural number. If $H$ is a proper subgroup of $G$, then $H$ is a properly contained in $N(H)$; hence, if $H$ is a subgroup of order $p^{n-1}$, then $H$ is normal in $G$".
and the author starts the proof by saying: " Let $K$ be a maximal normal group of $G$ contained in $H$......"
Well how do you know $H$ contains a maximal normal group of $G$? This seems nontrivial to me......any help will be appreciated.

This is an understandable misreading of the statement. The meaning of

Let $K$ be a maximal normal subgroup of $G$ contained in $H$

is: Let $K$ be a normal subgroup of $G$ contained in $H$, which is maximal with respect to this property. Informally, "let $K$ be one of the largest among the normal subgroups of $G$ that are contained in $H$". More formally: If $\mathcal S$ denotes the collection of all subgroups of $H$ that are normal in $G$, let $K$ an element of $\mathcal S$ that is maximal with respect to inclusion.
