Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

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

Contents 1 Statement 2 Related facts 2.1 Generalizations 3 Facts used 4 Proof

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 such that both the kernel of ρ and ρ ^{− 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 such that the kernel of ρ and ρ ^{− 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 , 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 for this action, with the quotient map.
3. V (the kernel of ρ) 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. : 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, . Since the action is trivial, .

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