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1.
Francesco G. Russo 《Quaestiones Mathematicae》2016,39(8):1019-1036
We show some results on the probability that a randomly picked pair (H, K) of subgroups of a finite group G satisfies [H, K] = 1. This notion of probability is related with the subgroup S-commutativity degree in [D.E. Otera and F.G. Russo, Subgroup S-commutativity degree of finite groups, Bull. Belgian Math. Soc. 19 (2012), 373–382]. Numerical inequalities will be illustrated, in order to find connections with the subgroup commutativity degree and with the commutativity degree of G. On the other hand, the literature has known a significant growth in the last years and so we discuss a series of new open problems, which are arousing interest in the area. The presence of a large bibliography allows us to have a detailed overview of the status of knowledge on the topic. 相似文献
2.
A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G. 相似文献
3.
A subgroup H of G is said to be $\pi$-quasinormal in G if it
permute with every Sylow subgroup of G.
In this paper, we extend the study on the structure of a finite group under the
assumption that some subgroups of G are
$\pi$-quasinormal in G. The main result we proved
in this paper is the following:Theorem 3.4.
Let ${\cal F}$ be a saturated formation
containing the supersolvable groups. Suppose that G is a group with a
normal subgroup H such that $G/H \in {\cal F}$,
and all maximal subgroups of any Sylow subgroup of $F^{*}(H)$ are $\pi$-quasinormal in
G, then $G \in {\cal F}$.
Received: 10 May 2002 相似文献
4.
Finite groups G=AB factorized by two subgroups A and B such that every subgroup of A permutes with every subgroup of B are studied in this paper. The behaviour of such products with respect to the class of finite groups in which Sylow-permutability is transitive is analyzed. 相似文献
6.
For a Hall system of a finite solvable group G, it is known that the set
of -permutable subgroups is a sublattice of the subgroup lattice of G. We investigate the class SPM of groups in which the lattice
is modular. We prove that if
is modular, then U V for all
(an evidently stronger condition). Both of these phenomena—the modularity of
and whether two -permutable subgroups U and V permute with each other—are shown to be determined locally, by what happens at each prime. The class SPM is shown to be quotient closed, but not direct product or subgroup closed.This revised electronic version of the Abstract includes the formulas that were missing in the previous electronic version published online in September 2004. 相似文献
7.
A subgroup H of a finite group
G is called c-normal in
G if there exists a normal subgroup
N of G such that
G = HN and $H \cap N \leq H_{G} = {\rm core}_{G}(H)$. In this paper, we investigate the class of groups
of which every maximal subgroup of its Sylow
p-subgroup is c-normal and the
class of groups of which some minimal subgroups of its Sylow
p-subgroup is c-normal for some prime number
p. Some interesting results are obtained and
consequently, many known results related to
p-nilpotent groups and
p-supersolvable groups are generalized. 相似文献
8.
Papiya Bhattacharjee 《Quaestiones Mathematicae》2018,41(1):81-98
The article introduces a new class of lattice-ordered groups. An ?-group G is lamron if Min(G)?1 is a Hausdorff topological space, where Min(G)?1 is the space of all minimal prime subgroups of G endowed with the inverse topology. It will be evident that lamron ?-groups are related to ?-groups with stranded primes. In particular, it is shown that for a W-object (G,u), if every value of u contains a unique minimal prime subgroup, then G is a lamron ?-group; such a W-object will be said to have W-stranded primes. A diverse set of examples will be provided in order to distinguish between the notions of lamron, stranded primes, W-stranded primes, complemented, and weakly complemented ?-groups. 相似文献
9.
Francesco de Giovanni Alessio Russo Giovanni Vincenzi 《Mediterranean Journal of Mathematics》2007,4(1):65-71
A subgroup X of a group G is called pronormal-by-finite if there exists a pronormal subgroup Y of G such that Y ≤ X and |X : Y| is finite. The structure of (generalized) soluble groups in which all subgroups are pronormal-by-finite is investigated.
Among other results, it is proved in particular that a finitely generated soluble group with such property is central-by-finite,
provided that it has no infinite dihedral sections. 相似文献
10.
Daniel T. Wise 《Inventiones Mathematicae》2002,149(3):579-617
A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M
g
∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal
subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and
at least 4 sides, then G is residually finite.
Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002 相似文献
11.
László Héthelyi 《Monatshefte für Mathematik》2000,130(3):201-209
In this paper we investigate the action of a p-group G on its powerful normal subgroup N. We shall mainly be concerned to find conditions which guarantee that G acts uniserially on N. We shall also study what consequences uniserial action on N has on the structure of N.
(Received 21 August 1998; in revised form 28 June 1999) 相似文献
12.
Nikolai Gordeev 《Journal of Pure and Applied Algebra》2009,213(2):250-258
We obtain the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g∈G such that for any 3 elements a1,a2,a3∈G the subgroup generated by the elements , i=1,2,3, is solvable. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 4 elements generate a solvable subgroup. The latter result also follows from a theorem of P. Flavell on {2,3}′-elements in the solvable radical of a finite group (which does not use the classification of finite simple groups). 相似文献
13.
Pavel Shumyatsky 《Monatshefte für Mathematik》2007,152(2):169-175
The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x
1,n
y
1] ⋯ [x
m,n
y
m
])q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite. 相似文献
14.
A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G=HK and H∩K=1. In this paper, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in NG(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained. 相似文献
15.
Françcoise Point 《Archive for Mathematical Logic》2001,40(7):525-529
We show that if G is a group of finite Morley rank, then the verbal subgroup <w(G)> is of finite width, where w is a concise word. As a byproduct, we show that if G is any abelian-by-finite group, then G
n
=<x
n
(G)> is definable.
Received: 15 March 1996 / Published online: 18 July 2001 相似文献
16.
In a finite group G every element can be factorized in such a way that there is one factor for each prime divisor p of | G |, and the order of this factor is pα for some integer α ≧ 0. We define g ∈G to be uniquely factorizable if it has just one such factorization (whose factors must be pairwise commuting). We consider the existence of uniquely factorizable
elements and its relation to the solvability of the group. We prove that G is solvable if and only if the set of all uniquely factorizable elements of G is the Fitting subgroup of G. We also prove various sufficient conditions for the non-existence of uniquely factorizable elements in non-solvable groups.
Received: 9 June 2005 相似文献
17.
Let H and K be normal subgroups of a finite group G and let K≤H. If A is a subgroup of G such that AH=AK or A∩H=A∩K, we say that A covers or avoids H/K respectively. The purpose of this paper is to investigate factor groups of a finite group G using this concept. We get some characterizations of a finite group being solvable or supersolvable and generalize some known results. 相似文献
18.
Let G be a finite group with no chief factor simple of Lie type E
8(q) and C a cyclic subgroup of largest order in G. It is shown that at most two primes in the open interval ([|C|/2], |C|) divide |G|.Received: 14 January 2005 相似文献
19.
We prove conditions for a product of distinct subgroups of an arbitrary group G to be a subgroup of G. In particular, the normal closure of any A ≤ G is equal to the product of some distinct conjugates of A. As an application of the later result we derive constraints on the size of a nontrivial conjugacy class of a finite non-Abelian simple group. 相似文献
20.
Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]i, a,b,c,d∈G and i?0, then the entire derived subgroup G′ satisfies a positive law. In fact, G′ is an extension of a nilpotent group by a locally finite group of finite exponent. 相似文献