Abnormal normalizer and 2-subnormal not implies normal

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

• 2-subnormal not implies hypernormalized
• Abnormal normalizer not implies pronormal

Other related facts

• Normalizer of 2-subnormal subgroup may have arbitrarily large subnormal depth

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 $G$.