Pronormal 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 G.

Related facts

Stronger facts

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).

Pronormal subgroup

Further information: Pronormal subgroup

A subgroup H of a group G is termed paranormal in G if for any g \in G, there exists x \in \langle H, H^g \rangle such that H^x = H^g.

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. Pronormal implies intermediately subnormal-to-normal
  2. Nilpotent implies every subgroup is subnormal
  3. Normality satisfies intermediate subgroup condition

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.

Now suppose g \in G is such that H^g \le K. Then, \langle H, H^g \rangle \le K. By fact (3), H is normal in \langle H, H^g \rangle. Thus, for any x \in \langle H, H^g \rangle, we have H^x \le H.

By the definition of pronormality, we also have x \in \langle H, H^g \rangle such that H^x = H^g \rangle. Since H^x \le H, we get H^g \le H, completing the proof.

References

Textbook references