On subgroups that are nearly pronormal

Abstract

A subgroup $$$H$$$ is called abnormal in the group $$$G$$$ if $$$g \in {<}H, H^g{>}$$$ for each $$$g \in G$$$. A subgroup $$$H$$$ of a group $$$G$$$ is called a strong selfnormalizer if for all subgroups $$$K$$$ such that $$$H \leqslant K$$$ the equality $$$N_K(H) = H$$$ is true. We obtain the examples of non-abnormal but strong selfnormalizer subgroups.