Prime power order implies no proper nontrivial characteristic direct factor
Suppose is a group of prime power order. Then 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.
Given: A group of prime power order, expressed as an internal direct product of nontrivial subgroups and .
To prove: is not characteristic in .
Proof: Let be a maximal subgroup of . Then, is normal and is cyclic of prime order. Let be a subgroup of order contained in the center of . Thus, we can construct a surjective homomorphism with kernel .
Now, consider the map:
This map is clearly a bijection from to . It is a homomorphism because . Thus, is an automorphism of . Moreover, does not leave invariant, showing that is not characteristic.