Prime power order implies no proper nontrivial characteristic direct factor

Statement

Suppose $P$ is a group of prime power order. Then $P$ has no proper nontrivial characteristic direct factor. In other words, there is no proper nontrivial characteristic subgroup that is also a factor in a direct product.

Facts used

1. Nilpotent implies every maximal subgroup is normal

Proof

Given: A group $P$ of prime power order, expressed as an internal direct product of nontrivial subgroups $H$ and $K$.

To prove: $H$ is not characteristic in $P$.

Proof: Let $N$ be a maximal subgroup of $H$. Then, $N$ is normal and $H/N$ is cyclic of prime order. Let $Q$ be a subgroup of order $p$ contained in the center of $K$. Thus, we can construct a surjective homomorphism $\alpha:H \to Q$ with kernel $N$.

Now, consider the map: $\sigma: (h,k) \mapsto (h,\alpha(h)k)$.

This map is clearly a bijection from $P$ to $P$. It is a homomorphism because $\alpha(h) \in Z(K)$. Thus, $\sigma$ is an automorphism of $P$. Moreover, $\sigma$ does not leave $H$ invariant, showing that $H$ is not characteristic.