Groupprops, The Group Properties Wiki (pre-alpha)

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 1.1 Statement with symbols 2 Related facts 2.1 Similar facts 3 Facts used 4 Proof

Statement

Statement with symbols

Suppose G is a group and p is a prime number such that G has no nontrivial abelian normal p-subgroup. Suppose H is a p-divisible normal subgroup of G: in other words, every element of H is the p^th power of some element of H. Then, H is a potentially characteristic subgroup of G: there exists a group K containing G such that H is characteristic in K.

Related facts

Similar facts

• Finite NPC theorem
• Finite NIPC theorem
• No nontrivial abelian normal p-subgroup implies every normal subgroup is strongly image-potentially characteristic

Facts used

1. Cayley's theorem
2. Characteristicity is centralizer-closed
3. Quotient group acts on abelian normal subgroup
4. Characteristicity is transitive

Proof

Given: A finite group G, a prime p such that no element of G has order p. A normal subgroup H of G such that every element of H has a p^th root.

To prove: There exists a group K containing G such that H is characteristic in K.

Proof:

1. Let L = G / H. By fact (1), L is a subgroup of the symmetric group , which in turn can be embedded in the automorphism group of a vector space over the field of p elements (the dimension of the vector space is | L | ). Thus, L has a faithful representation on a vector space V 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: V is an abelian normal p-subgroup. By assumption, has no nontrivial abelian normal p-subgroup, so V is the unique largest abelian normal p-subgroup. Hence, V is a characteristic subgroup of K.
4. C_K(V) is characteristic in K: This follows from the previous step and fact (2).
5. : Since V is abelian, the quotient group acts on V (fact (3)); 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, by the assumption that every element of H is a p^th power, H is precisely the set of p^th powers 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 (4), H is characteristic in K.