Subgroup property between normal and subnormal-to-normal is not transitive

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Statement

Suppose p is a subgroup property that is weaker than the property of being a Normal subgroup (?), but is stronger than the property of being a Subnormal-to-normal subgroup (?): in other words, every subnormal subgroup satisfying p is normal. Then, p is not a Transitive subgroup property (?).

Particular cases

Facts used

  1. Normality is not transitive

Proof

Since normality is not transitive (fact (1)), we can construct groups H \le K \le G such that H is normal in K and K is normal in G, but H is not normal in G. We then have:

  • H satisfies property p in K and K satisfies property p in G: This follows because p is weaker than normality.
  • H does not satisfy property p in G: By construction, H is subnormal in G, so if H satisfies property p in G, H is normal in G, contradicting our assumption.

Thus, p is not transitive.

This shows that any example of normality not being transitive yields an example of p not being transitive.