Kernel of a characteristic action on an abelian group with which it is characteristic in the direct product implies potentially characteristic

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

Statement

Suppose $H$ is a subgroup of a group $G$ . Suppose there exists an abelian group $V$ satisfying the following two conditions:

There is a homomorphism $\alpha:G \to \operatorname{Aut}(V)$ with kernel $H$ , such that $V$ is a characteristic subgroup of the semidirect product $V \rtimes G$ .
$H$ is a characteristic subgroup of $V \times H$ .

Then, $H$ is a potentially characteristic subgroup of $G$ . In fact, $H$ is a characteristic subgroup of $V \rtimes G$ .

Facts used

Proof

Given: A subgroup $H \le G$ . An abelian group $V$ and a homomorphism $\alpha: G \to \operatorname{Aut}(V)$ with kernel $H$ . $V$ is characteristic in $V \rtimes G$ , and $H$ is characteristic in $V \times H$ .

To prove: $H$ is characteristic in the semidirect product $K = V \rtimes G$ .

Proof:

$V$ is characteristic in $K$ : This is by assumption.
$C K (V)$ is characteristic in $K$ : This follows from fact (1).
$C_K(V) = V \times H$ : Since $V$ is abelian, the quotient group $K/V \cong G$ acts on $V$ (fact (2)); 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, $C_K(V) = V \times H$ .
$H$ is characteristic in $C_K(V) = V \times H$ : This is again by assumption.
$H$ is characteristic in $K$ : This follows from steps (2) and (4) and fact (3).

Facts about Kernel of a characteristic action on an abelian group with which it is characteristic in the direct product implies potentially characteristicRDF feed

Fact related to	NPC conjecture +
Uses	Characteristicity is centralizer-closed +, Quotient group acts on abelian normal subgroup +, and Characteristicity is transitive +

Kernel of a characteristic action on an abelian group with which it is characteristic in the direct product implies potentially characteristic

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