The aim of the study is to characterize all finite groups
that satisfy the normalizer conditions stated in this manuscript. Group
and character theoretic methods are used in the study and it is proved
that such groups are not simple. Specifically, the following result is
established. Let a finite group G have a maximal subgroup H satisfying:
(I)H = XP<t>, where P = <x, y: x3 = y3
= [x, y] = 1> and t2 = (zt)2 = 1 for all z in
H. (II) X = RxKxT, where R has odd order and y acts fixed-point-free on
X; K and T are 2-groups, xy centralizes K and acts fixed-pint-free on
T; x centralizes T and acts fixed-point-free on K. T≠1, K≠1.
(III) H is the only maximal subgroup of G containing XP and |Ω1(Z(KxT))|>4.
Then G is not a simple group.