Groupprops, The Group Properties Wiki (pre-alpha)

TIP: Read more about how the definition in Groupprops is structured

ABOUT US: Read our purpose statement and learn what makes us special

ALSO CHECK OUT: Topospaces: The Topology Wiki

From Groupprops

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

Contents 1 Statement 2 Related facts 2.1 Related facts about potentially characteristic subgroups with similar proofs 2.2 Other related facts about potentially characteristic subgroups 2.3 Analogous facts for image-potentially characteristic subgroups 2.4 Breakdown of stronger facts 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 containing G such that H is a characteristic subgroup of K.

Related facts

Related facts about potentially characteristic subgroups with similar proofs

• No nontrivial abelian normal p-subgroup for some prime p implies every p-divisible normal subgroup is potentially characteristic
• Kernel of a characteristic action on an abelian group with which it is characteristic in the direct product implies potentially characteristic

Other related facts about potentially characteristic subgroups

• Finite normal implies potentially characteristic
• Periodic normal implies potentially characteristic
• Central implies potentially characteristic
• Normal subgroup contained in hypercenter is potentially characteristic
• Abelian implies every subgroup is potentially characteristic
• Nilpotent implies every normal subgroup is potentially characteristic
• Central implies potentially verbal in finite

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 semi-strongly potentially characteristic: If H is a normal subgroup of a finite group K, it is not necessary that there exists a group G containing K as a normal subgroup and H as a characteristic subgroup.

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 , 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.
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. : Since V is abelian, the quotient group 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, .
6. H is characteristic in : H is a normal subgroup of , on account of being a direct factor. Further, it is a normal p'-Hall subgroup, so by fact (2), it is characteristic in .
7. H is characteristic in K: By steps (4) and (5), is characteristic in K, and by step (6), H is characteristic in . Thus, by fact (5), H is characteristic in K.