# Finite NPC theorem

From Groupprops

## Contents

## Statement

Suppose is a finite group and is a normal subgroup of . Then, there exists a finite group containing such that is a characteristic subgroup of .

## 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).

### 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 is a normal subgroup of a finite group , it is
*not*necessary that there exists a group containing as a normal subgroup and as a characteristic subgroup.

## Facts used

- Cayley's theorem
- Normal Hall implies characteristic
- Characteristicity is centralizer-closed
- Quotient group acts on abelian normal subgroup
- Characteristicity is transitive

## Proof

**Given**: A finite group , a normal subgroup of .

**To prove**: There exists a group containing such that is characteristic in .

**Proof**:

- Let . Suppose is a prime not dividing the order of . By fact (1), is a subgroup of the symmetric group , which in turn can be embedded in the general linear group where . Thus, has a faithful representation on a vector space of dimension over the prime field of order .
- Since , a faithful representation of on gives a representation of on whose kernel is . Let be the semidirect product for this action. We can also think of as a wreath product of the group of prime order by for this action.
- is characteristic in : In fact, is a normal -Sylow subgroup, and hence is characteristic (fact (2)) (it can be defined as the set of all elements whose order is a power of ).
- is characteristic in : This follows from the previous step and fact (3).
- : Since is abelian, the quotient group acts on (fact (4)); in particular, any two elements in the same coset of have the same action by conjugation on . Thus, the centralizer of comprises those cosets of for which the corresponding element of fixes . This is precisely the cosets of elements of . Thus, . Since the action is trivial, .
- is characteristic in : is a normal subgroup of , on account of being a direct factor. Further, it is a normal -Hall subgroup, so by fact (2), it is characteristic in .
- is characteristic in : By steps (4) and (5), is characteristic in , and by step (6), is characteristic in . Thus, by fact (5), is characteristic in .