Finite NPC theorem

Statement
Suppose $$G$$ is a finite group and $$H$$ is a normal subgroup of $$G$$. Then, there exists a finite group $$K$$ containing $$G$$ such that $$H$$ is a characteristic subgroup of $$K$$.

Related facts about potentially characteristic subgroups with similar proofs

 * Normal equals potentially characteristic: The general version of the result.
 * Finite NIPC theorem: An analogous results for quotients/images (finite group version).
 * Normal equals image-potentially characteristic: An analogous results for quotients/images (general version).

Other related facts about potentially characteristic subgroups

 * Central implies potentially verbal in finite

Analogous facts for image-potentially characteristic subgroups

 * Finite NIPC theorem: Analogous statement for images; the proof uses the same construction.

Breakdown of stronger facts

 * Normal not implies normal-extensible automorphism-invariant in finite
 * Normal not implies normal-potentially characteristic: If $$H$$ is a normal subgroup of a finite group $$G$$, it is not necessary that there exists a group $$K$$ containing $$G$$ as a normal subgroup and $$H$$ as a characteristic subgroup.

Facts used

 * 1) uses::Cayley's theorem
 * 2) uses::Normal Hall implies characteristic
 * 3) uses::Characteristicity is centralizer-closed
 * 4) uses::Quotient group acts on abelian normal subgroup
 * 5) uses::Characteristicity is transitive

Proof
Given: A finite group $$G$$, a normal subgroup $$H$$ of $$G$$.

To prove: There exists a group $$K$$ containing $$G$$ such that $$H$$ is characteristic in $$K$$.

Proof:


 * 1) Let $$L = G/H$$. Suppose $$p$$ is a prime not dividing the order of $$G$$. By fact (1), $$L$$ is a subgroup of the symmetric group $$\operatorname{Sym}(L)$$, which in turn can be embedded in the general linear group $$GL(n,p)$$ where $$n = |L|$$. Thus, $$L$$ has a faithful representation on a vector space $$V$$ of dimension $$n$$ over the prime field of order $$p$$.
 * 2) Since $$L = G/H$$, a faithful representation of $$L$$ on $$V$$ gives a representation of $$G$$ on $$V$$ whose kernel is $$H$$. Let $$K$$ be the semidirect product $$V \rtimes G$$ for this action.
 * 3) $$V$$ is characteristic in $$K$$: In fact, $$V$$ is a normal $$p$$-Sylow subgroup, and hence is characteristic (fact (2)) (it can be defined as the set of all elements whose order is a power of $$p$$).
 * 4) $$C_K(V)$$ is characteristic in $$K$$: This follows from the previous step and fact (3).
 * 5) $$C_K(V) = V \times H$$: Since $$V$$ is abelian, the quotient group $$K/V \cong G$$ acts on $$V$$ (fact (4)); in particular, any two elements in the same coset of $$V$$ have the same action by conjugation on $$V$$. Thus, the centralizer of $$V$$ comprises those cosets of $$V$$ for which the corresponding element of $$G$$ fixes $$V$$. This is precisely the cosets of elements of $$H$$. Thus, $$C_K(V) = V \rtimes H$$. Since the action is trivial, $$C_K(V) = V \times H$$.
 * 6) $$H$$ is characteristic in $$V \times H$$: $$H$$ is a normal subgroup of $$V \times H$$, on account of being a direct factor. Further, it is a normal $$p'$$-Hall subgroup, so by fact (2), it is characteristic in $$V \times H$$.
 * 7) $$H$$ is characteristic in $$K$$: By steps (4) and (5), $$V \times H$$ is characteristic in $$K$$, and by step (6), $$H$$ is characteristic in $$V \times H$$. Thus, by fact (5), $$H$$ is characteristic in $$K$$.