Finite Groups With Maximal Normalizers I

TL;DR

This paper studies specific finite p-groups where non-normal subgroups have maximal normalizers, revealing bounds on the center's index depending on whether p is 2 or odd.

Contribution

It characterizes p-groups with maximal normalizers for non-normal subgroups and establishes bounds on the index of their centers.

Findings

For p=2, the center's index is at most 16.

For odd p, the center's index is at most p^3.

Provides structural insights into these special p-groups.

Abstract

We examine $p$-groups with the property that every non-normal subgroup has a normalizer which is a maximal subgroup. In particular we show that for such a $p$-group $G$, when $p=2$, the center of $G$ has index at most 16 and when $p$ is odd the center of $G$ has index at most $p^3$.