Abstract
The groups having exactly one normalizer are Dedekind groups. All finite groups with exactly two normalizers were classified by Pérez-Ramos in 1988. In this paper we prove that every finite group with at most 26 normalizers of {2,3,5}-subgroups is soluble and we also show that every finite group with at most 21 normalizers of cyclic {2,3,5}-subgroups is soluble. These confirm Conjecture 3.7 of Zarrin (Bull Aust Math Soc 86:416–423, 2012).
Highlights
Introduction and notationAll groups considered in this paper are finite
For example in [2] it was shown that an insoluble group has exactly 22 non-nilpotent subgroups if and only if it is isomorphic to A5 or SL(2, 5) and that confirmed a conjecture of Zarrin [17]
We say that a group G is an Nn-group (Ncn-group, respectively) if it has exactly n normalizers of subgroups
Summary
Introduction and notationAll groups considered in this paper are finite. We use conventional notions and notations. ([18, Conjecture 3.7]) Every Ncn-group with n 21 is soluble. We generalize these conjectures in the class of finite groups and we show that it is enough to consider only normalizers of {2, 3, 5}subgroups of a group.
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