Direct factor implies transitively normal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., direct factor) must also satisfy the second subgroup property (i.e., transitively normal subgroup)
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Statement

Suppose H is a direct factor of a group G. Then, H is a transitively normal subgroup of G. In other words, for any normal subgroup K of H, K is also normal in G.

Facts used

  1. Direct factor implies central factor
  2. Central factor implies transitively normal

Proof

Proof using given facts

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