Groupprops, The Group Properties Wiki (pre-alpha)

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From Groupprops

Statement

Suppose p is a group property such that there is a nontrivial group satisfying pband p is closed under taking extensions (in other words, if a normal subgroup and its quotient group both satisfy p, so does the whole group). Then, there exists a group G satisfying p, a normal subgroup K of G satisfying p, and a normal subgroup H of K, also satisfying p, that is not normal in G.

In fact, for any nontrivial group H satisfying p, we can find K and G so that the above occurs.

Related facts

• Normality is not transitive
• Conjunction of normality with any nontrivial finite-direct product-closed group property is not transitive
• Every nontrivial normal subgroup is potentially normal-and-not-2-subnormal