Abnormal normalizer and 2-subnormal not implies normal

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Statement

A 2-subnormal subgroup (?) with abnormal normalizer, i.e., a 2-subnormal subgroup whose normalizer in the whole group is an abnormal subgroup, need not be a normal subgroup.

Related facts

Corollaries

Other related facts

Proof

An example of the symmetric group on four letters

Further information: symmetric group:S4

Let G = S_4 by the symmetric group on the set \{ 1,2,3,4\}, and H be the two-element subgroup generated by the double transposition (1,2)(3,4). Then, the normalizer N_G(H) is the dihedral group of order eight, which is a maximal non-normal subgroup, and hence abnormal. Hence, H has abnormal normalizer.

On the other hand, H is contained in the subgroup K of order four, comprising the identity and the double transpositions. H is normal in K, and K is normal in G, so H is 2-subnormal in G.