Normal core of normalizer

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This article defines a subgroup operator related to the subgroup property normal subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

Definition

The normal core of normalizer of a subgroup H of a group G is defined in the following equivalent ways:

  1. It is the largest normal subgroup of G that normalizes H. In other words, it is the largest normal subgroup of G contained in the normalizer N_G(H).
  2. It is the normal core of the normalizer N_G(H) in G.

H is contained in the normal core of normalizer of H if and only if H is a 2-subnormal subgroup of G. In this case, if K is the normal core of normalizer of H, then the ascending chain H \le K \le G is the unique fastest ascending subnormal series for H in G. Further information: 2-subnormal subgroup has a unique fastest ascending subnormal series