WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., WNSCDIN-subgroup) must also satisfy the second subgroup property (i.e., subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed)
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Statement

If K is a WNSCDIN-subgroup of G, and H is a normalizer-relatively normal conjugation-invariantly relatively normal subgroup of K with respect to G, then H is a weakly closed subgroup of K with respect to G.

Definitions used

WNSCDIN-subgroup

Further information: WNSCDIN-subgroup

A subgroup H of a group G is a WNSCDIN-subgroup if whenever A,B \subseteq H are normal subsets of H and g \in G is such that gAg^{-1} = B, there exists k \in N_G(H) such that kAk^{-1} = B.

Normalizer-relatively normal subgroup

Further information: Normalizer-relatively normal subgroup

Suppose H \le K \le G. H is normalizer-relatively normal in K with respect to G if H is normal in N_G(K). In other words, N_G(K) \le N_G(H).

Conjugation-invariantly relatively normal subgroup

Further information: Conjugation-invariantly relatively normal subgroup

Suppose H \le K \le G. H is conjugation-invariantly relatively normal in K with respect to G if H is normal in every conjugate of K in G that contains H.

Weakly closed subgroup

Further information: Weakly closed subgroup

Suppose H \le K \le G. H is weakly closed in K relative to G if, for any x \in G such that xHx^{-1} \le K, we have xHx^{-1} \le H.

Related facts

Converse

Applications

Proof

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Given: Groups H \le K \le G such that K is a WNSCDIN-subgroup of G. H is normal in N_G(K), and H is normal in gKg^{-1} for all g \in G such that H \le gKg^{-1}. We have x \in G such that xHx^{-1} \le K.

To prove: xHx^{-1} \le H (in fact, we'll prove equality).

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 H is a normal subgroup of x^{-1}Kx xHx^{-1} \le K Since xHx^{-1} \le K, conjugating both sides by x^{-1} yields H \le x^{-1}Kx. By the assumption that H is normal in every conjugate of K containing it, H is a normal subgroup of x^{-1}Kx.
2 xHx^{-1} is a normal subgroup of K. Step (1) This follows from the previous step, and conjugating by K.
3 H is a normal subgroup of K. normality satisfies intermediate subgroup condition H \le K, H is normal in N_G(K) Step-fact direct
4 H and xHx^{-1} are conjugate in N_G(K) K is a WNSCDIN-subgroup of G Steps (2), (3) Given-step direct
5 H = xHx^{-1} H is normal in N_G(K) Step (4) Given-step direct