Pronormality is not transitive

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This article gives the statement, and possibly proof, of a subgroup property (i.e., pronormal subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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Statement

Verbal statement

A pronormal subgroup of a pronormal subgroup need not be pronormal in the whole group.

Statement with symbols

It is possible to have a group G with subgroups H \le K \le G such that H is pronormal in K and K is pronormal in G, but H is not pronormal in G.

Related facts

Generalization and other instances

Other instances of the generalization include:

Facts used

  1. Normal implies pronormal
  2. Pronormal and subnormal implies normal
  3. Normality is not transitive

Proof

By fact (3), construct subgroups H \le K \le G such that H is normal in K, K is normal in G, but H is not normal in G.

  • By fact (1), H is pronormal in K and K is pronormal in G.
  • By definition, H is subnormal in G, so by fact (2), if H were pronormal in G, H would also be normal in G. But by construction, H is not normal in G, so H is not pronormal in G.

In particular, any example showing that normality is not transitive also shows that pronormality is not transitive.