Groups whose non-normal subgroups have a transitive normality relation (original) (raw)
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A note on finite đ«đźđŻ-groups
Journal of Group Theory, 2007
A finite group G is said to be a PST-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a T *-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that finite PST-groups and finite T *-groups are one and the same. A new characterisation of soluble PST-groups is also presented.
Journal of Algebra, 1980
A well-known theorem of Wielandt states that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G. The structure of a nonnilpotent group, each of whose proper subgroups is nilpotent, has been analyzed by Schmidt and R6dei [5, Satz 5.1 and Satz 5.2, pp. 280-281]. In [1], Buckley investigated the structure of a PN-group (i.e., a finite group in which every minimal subgroup is normal), and proved (i) that a PN-group of odd order is supersolvable, and (ii) that certain factor groups of a PN-group of odd prime power order are also PN-groups. Earlier, Gaschiitz and It5 [5, Satz 5.7, p. 436] had proved that the commutator subgroup of a finite PN-group is p-nilpotent for each odd prime p. This paper is a sequel to [9] and our object here is to prove the following statement. THEOREM. If G is a finite nonPN-group, each of whose proper subgroups is a PN-group, then one of the following statements is true: (a) G is the dihedral group of order 8.
Structure of finite groups with trait of non-normal subgroups II
International Journal of Group Theory
A finite non-Dedekind group G is called an đ©đđ-group if all non-normal abelian subgroups are cyclic. In this paper, all finite đ©đđ-groups will be characterized. Also, it will be shown that the center of non-nilpotent đ©đđ- groups is cyclic. If đ©đđ-group G has a non-abelian non-normal Sylow subgroup of odd order, then other Sylow subgroups of G are cyclic or of quaternion type.
On Groups with All Subgroups Subnormal or Soluble of Bounded Derived Length
arXiv: Group Theory, 2012
In this paper, we deal with locally graded groups whose subgroups are either subnormal or soluble of bounded derived length, say d. In particular, we prove that every locally (soluble-by-finite) group with this property is either soluble or an extension of a soluble group of derived length at most d by a finite group, which fits between a minimal simple group and its automorphism group. We also classify all the finite non-abelian simple groups whose proper subgroups are metabelian.
Locally soluble groups with min-n
Journal of the Australian Mathematical Society, 1974
It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of c...
The structure of finite groups with trait of non-normal subgroups
Communications in Algebra
In this paper, we investigate the finite groups all of whose non-normal nilpotent subgroups are cyclic. We show that such groups are solvable with cyclic centers. If G is a non-supersolvable group, then G has only one non-cyclic Sylow subgroup which is either isomorphic to Q8 or is of type (q, q).
On certain minimal non-Y-groups for some classes Y
2015
Let {Ξn} â n=1 be a sequence of words. If there exists a positive integer n such that Ξm(G) = 1 for every m â„ n , then we say that G satisfies (*) and denote the class of all groups satisfying (*) by X {Ξn} â n=1. If for every proper subgroup K of G , K â X {Ξn} â n=1 but G / â X {Ξn} â n=1 , then we call G a minimal non-X {Ξn} â n=1-group. Assume that G is an infinite locally finite group with trivial center and Ξi(G) = G for all i â„ 1. In this case we mainly prove that there exists a positive integer t such that for every proper normal subgroup N of G , either Ξt(N) = 1 or Ξt(CG(N)) = 1. We also give certain useful applications of the main result.
Soluble Groups with Extremal Conditions on Commutators
Ricerche di Matematica
In this article we investigate the structure of soluble groups G such that [H, K] = H â© K for all proper distinct normal subgroups H and K of G. Moreover, also the class consisting of all soluble groups G such that [H, K] = {1} for all proper distinct normal subgroups H and K of G is considered.