A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
Solution 1:
Hint: Apply Cauchy's theorem to $C(G/N_k)$. This will give you a normal subgroup of $G$ of order $p^{k+1}$.
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
Solution 1:
Hint: Apply Cauchy's theorem to $C(G/N_k)$. This will give you a normal subgroup of $G$ of order $p^{k+1}$.