Finite NIPC theorem

This fact is related to: NIPC conjecture
View other facts related to NIPC conjecture | View terms related to NIPC conjecture

Statement

Suppose $G$ is a finite group and $H$ is a normal subgroup of $G$. Then, there exists a finite group $K$ and a surjective homomorphism $\rho :K\to G$ such that both the kernel of $\rho$ and $\rho ^{-1}(H)$ are characteristic subgroups of $K$.

Related facts

• Finite NPC theorem
• No nontrivial abelian normal p-subgroup for some prime p implies every normal subgroup is strongly image-potentially characteristic

Generalizations

• Kernel of a characteristic action on an abelian group implies strongly image-potentially characteristic

Facts used

1. Cayley's theorem
2. Normal Hall implies characteristic
3. Characteristicity is centralizer-closed
4. Quotient group acts on abelian normal subgroup

Proof

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

To prove: There exists a group $K$ and a surjective homomorphism $\rho :K\to G$ such that the kernel of $\rho$ and $\rho ^{-1}(H)$ are both 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, with $\rho :K\to G$ the quotient map.
3. $V$ (the kernel of $\rho$) 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. $C_{K}(V)=V\times H=\rho ^{-1}(H)$: Since $V$ is abelian, the quotient group $K/V$ 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, $C_{K}(V)=V\times H=\rho ^{-1}(H)$.

The last two steps show that $\rho ^{-1}(H)$ is characteristic in $K$, while step (3) shows that the kernel of $\rho$ is characteristic in $K$. This completes the proof.