Prime power order implies no proper nontrivial characteristic direct factor

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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.