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

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. Cayley's theorem
2. Normal Hall implies characteristic
3. Characteristicity is centralizer-closed
4. Quotient group acts on abelian normal subgroup
5. 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. We can also think of $K$ as a wreath product of the group of prime order $p$ by $G$ 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 $K$.