Weakly normal implies weakly closed in intermediate nilpotent

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Statement

Statement with symbols

Suppose H \le K \le G are groups such that:

Then, H is a Weakly closed subgroup (?) of K.

Related facts

Corollaries

Definitions used

For these definitions, H^g = g^{-1}Hg denotes the conjugate subgroup by g \in G. (This is the right-action convention; however, adopting a left-action convention does not alter any of the proof details).

Paranormal subgroup

Further information: Weakly normal subgroup

A subgroup H of a group G is termed paranormal in G if for any g \in G, H^g \le N_G(H) implies H^g \le H. In other words, it is weakly closed in its normalizer.

Weakly closed subgroup

Further information: Weakly closed subgroup

Suppose H \le K \le G are groups. We say H is weakly closed in K with respect to G if, for any g \in G such that H^g \le K, we have H^g \le H.

Facts used

  1. Weakly normal implies intermediately subnormal-to-normal
  2. Nilpotent implies every subgroup is subnormal

Proof

Given: H \le K \le G with H a paranormal subgroup of G and K a nilpotent group.

To prove: For any g \in G such that H^g \le K, we have H^g \le H.

Proof: By fact (2), H is a subnormal subgroup of K. By fact (1), H is therefore normal in K. Thus, K \le N_G(H).

Since H is weakly closed in N_G(H) by definition, whenever H^g \le N_G(H), we get H^g \le H. In particular, whenever H^g \le K, we get H^g \le H, so H is weakly closed in K.