Characteristic central factor of WNSCDIN implies WNSCDIN

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Characteristic central factor (?) and WNSCDIN-subgroup (?)), to another known subgroup property (i.e., WNSCDIN-subgroup (?))
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Statement

Suppose $H \leq K \leq G$ are groups. Suppose $H$ is a characteristic central factor of $K$ : in other words, $H$ is both a Characteristic subgroup (?) and a Central factor (?) of $K$ . Suppose $K$ is a WNSCDIN-subgroup of $G$ . Then, $H$ is also a WNSCDIN-subgroup of $G$ .

Facts used

Characteristic of normal implies normal

Proof

Given: A group $G$ , a WNSCDIN-subgroup $K$ of $G$ . A characteristic central factor $H$ of $K$ .

To prove: $H$ is a WNSCDIN-subgroup of $G$ . In other words, if $A$ and $B$ are normal subsets of $H$ that are conjugate in $G$ , then $A$ and $B$ are conjugate in $N_{G} (H)$ .

Proof:

$A, B$ are normal subsets of $K$ : Any inner automorphism of $K$ restricts to an inner automorphism of $H$ . In particular, since $A$ is invariant under all inner automorphisms of $H$ , it is invariant under all inner automorphisms of $H$ . Thus, $A$ (and similarly $B$ ), are normal subsets of $K$ .
$A, B$ are conjugate in $N_{G} (K)$ : This follows from the previous step, and the fact that $K$ is WNSCDIN in $G$ .
$N_{G} (K) \leq N_{G} (H)$ : $H$ is characteristic in $K$ and $K$ is normal in $N_{G} (K)$ . Thus, by fact (1), $H$ is normal in $N_{G} (K)$ . Thus, $N_{G} (K) \leq N_{G} (H)$ .
$A, B$ are conjugate in $N_{G} (H)$ : This follows from the previous two steps.