Paranormal 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: Paranormal subgroup

A subgroup H of a group G is termed paranormal in G if for any g \in G, H is a contranormal subgroup of \langle H, H^g \rangle; in other words, the normal closure of H in \langle H, H^g \rangle is the whole of \langle H, H^g \rangle.

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. Paranormal implies weakly normal
  2. Weakly normal implies weakly closed in intermediate nilpotent

Proof

The proof follows directly from facts (1) and (2).