Collection of groups satisfying a strong normal replacement condition

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Definition

Suppose \mathcal{S} is a finite collection of finite p-groups, i.e., groups of prime power order where the prime is p. We say that \mathcal{S} satisfies a strong normal replacement condition if it satisfies the following equivalent conditions:

  1. For any finite p-group P that contains a subgroup H isomorphic to an element of \mathcal{S}, P contains a normal subgroup K, also isomorphic to an element of \mathcal{S}, such that K is contained in the normal closure of H in P.
  2. For any finite p-group P that contains a 2-subnormal subgroup H isomorphic to an element of \mathcal{S}, P contains a normal subgroup K, also isomorphic to an element of \mathcal{S}, such that K is contained in the normal closure of H in P.# For any finite p-group Q and normal subgroup P of Q, if there exists a subgroup of P isomorphic to an element of \mathcal{S}, there exists a subgroup of P that is normal in Q and is isomorphic to an element of \mathcal{S}.

Relation with other properties

Stronger properties

Weaker properties