Finite NPC theorem

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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

Other related facts about potentially characteristic subgroups

Analogous facts for image-potentially characteristic subgroups

Breakdown of stronger facts

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.