# Finite NIPC theorem

From Groupprops

This fact is related to: NIPC conjecture

View other facts related to NIPC conjectureView terms related to NIPC conjecture |

## Statement

Suppose is a finite group and is a normal subgroup of . Then, there exists a finite group and a surjective homomorphism such that both the kernel of and are characteristic subgroups of .

## 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

## Facts used

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

## Proof

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

**To prove**: There exists a group and a surjective homomorphism such that the kernel of and are both 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, with the quotient map.
- (the kernel of ) 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, .

The last two steps show that is characteristic in , while step (3) shows that the kernel of is characteristic in . This completes the proof.