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Solvability of factorized finite groups
Abstract
Using classification theorems of
simple groups, we give a proof of a conjecture on factorized finite groups which is an extension of a
well known theorem due to P. Hall.
... We refer to [8,28,9] for previous studies for the class L = N of finite nilpotent groups, and to [18,19,20,21] for L being the class of finite metanilpotent groups and other relevant classes of groups. For the class L = S of finite soluble groups, A. Carocca in [12] proved the solubility of a product of S-connected soluble subgroups, which provides a first extension of the above-mentioned theorem of Thompson for products of groups (see Corollary 2). ...
... It is clear that the Main Theorem extends the above-mentioned theorems of Thompson [38] and Carocca [12]. It also implies the theorem of Guralnick, Kunyavskiȋ, Plotkin and Shalev [26] stated above (with A = G, B = x ; note that x G S is a normal (soluble) subgroup of G by the Main Theorem). ...
... By the Main Theorem, [A, B] ≤ G S = 1. Therefore A and B are normal in G, whence A = 1 or B = 1 or N ≤ A ∩ B. But the latter implies N = N ′ ≤ [A, B] = 1, a contradiction.There is another corollary to the Main Theorem that generalizes a result of Carocca[12] on the solubiltity of S-connected products of finite soluble groups:Corollary 2. Let G = AB be a finite group with S-connected subgroups A, B. Then A S = A ∩ G S and B S = B ∩ G S .In particular, if A and B are soluble then G is soluble. ...
A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups with subgroups such that is soluble for all and . In this case, the group G is said to be an -connected product of the subgroups A and B for the class of all finite soluble groups. Our main theorem states that is -connected if and only if [A,B] is soluble. In the course of the proof we derive a result of own interest about independent primes regarding the soluble graph of almost simple groups.
... For the class S of finite soluble groups, Carocca [7] showed that S-connected products of finite soluble groups are soluble. Structure and properties of N -connected products of finite groups (N de-✩ Research supported by Proyecto MTM2007-68010-C03-03, Ministerio de Educación y Ciencia and FEDER, Spain. ...
... Proof. Notice that if condition (a) holds, then G is soluble by Carocca's result [7,Theorem]. Hence we may assume for the proof that G is soluble. ...
... , G n are pairwise permutable and pairwise S-connected soluble subgroups of G = G 1 · · · G n . Then G is soluble by Carocca's result [7,Theorem]. ...
For a non-empty class of groups C, two subgroups A and B of a group G are said to be C-connected if 〈a,b〉∈C for all a∈A and b∈B. Given two sets π and ρ of primes, SπSρ denotes the class of all finite soluble groups that are extensions of a normal π-subgroup by a ρ-group.It is shown that in a finite group G=AB, with A and B soluble subgroups, then A and B are SπSρ-connected if and only if Oρ(B) centralizes AOπ(G)/Oπ(G), Oρ(A) centralizes BOπ(G)/Oπ(G) and G∈Sπ∪ρ. Moreover, if in this situation A and B are in SπSρ, then G is in SπSρ.This result is then extended to a large family of saturated formations F, the so-called nilpotent-like Fitting formations of soluble groups, and to finite groups that are products of arbitrarily many pairwise permutable F-connected F-subgroups.
... They have obtained the following nice result: Assume that G = AB is the product of two supersoluble subgroups A and B. If every subgroup of A is permutable with every subgroup of B, then G is supersoluble. In addition, they have also generalized the above mentioned result of Baer by replacing the condition of normality of A, B in G and using the following weaker condition: A permutes with all subgroups of B and B permutes with all subgroups of A. Their results in [3] were further developed and applied by many authors (see, for example, [1] [5][6][7][8], [14], [19]). We also notice that O.H.Kegel has also obtained many elegant results for soluble groups and supersoluble groups by considering the products of their subgroups (see [16][17][18]). ...
... Corollary 4.2 [8]. Let p be a prime number. ...
Let H and T be subgroups of a group G. Then we call H conditionally permutable (or in brevity, c-permutable) with T in G if there exists an element x ∈ G such that HT x = T x H. If H is c-permutable with T in < H, T >, then we call H completely c-permutable with T in G. By using the above concepts, we will give some new criterions for the supersolubility of a finite group G = AB, where A and B are both supersoluble groups. In particular, we prove that a finite group G is supersoluble if and only if G = AB, where both A, B are nilpotent subgroups of the group G and B is completely c-permutable in G with every term in some chief series of A. We will also give some applications of our new criterions.
... They have obtained the following nice result: Assume that G = AB is the product of two supersoluble subgroups A and B. If every subgroup of A is permutable with every subgroup of B, then G is supersoluble. In addition, they have also generalized the above mentioned result of Baer by replacing the condition of normality of A, B in G and using the following weaker condition: A permutes with all subgroups of B and B permutes with all subgroups of A. Their results in [3] were further developed and applied by many authors (see, for example, [1] [5][6][7][8], [14], [19]). We also notice that O.H.Kegel has also obtained many elegant results for soluble groups and supersoluble groups by considering the products of their subgroups (see [16][17][18]). ...
... Corollary 4.2 [8]. Let p be a prime number. ...
Let H and T be subgroups of a group G. Then we call H conditionally permutable (or in brevity, c-permutable) with T in G if there exists an element x ∈ G such that HT x = T x H. If H is c-permutable with T in < H, T >, then we call H completely c-permutable with T in G. By using the above concepts, we will give some new criterions for the supersolubility of a finite group G = AB, where A and B are both supersoluble groups. In particular, we prove that a finite group G is supersoluble if and only if G = AB, where both A, B are nilpotent subgroups of the group G and B is completely c-permutable in G with every term in some chief series of A. We will also give some applications of our new criterions.
... They also prove an analogous result by considering K nilpotent instead of G (Theorem 3.2). Later, Carocca [4] presented extensions of the preceding result considering p-supersolubility instead of supersolubility. Following Carocca [4], we say that the subgroups H and K of a group G are mutually permutable if H commutes with every subgroup of K and K commutes with every subgroup of H. ...
... Later, Carocca [4] presented extensions of the preceding result considering p-supersolubility instead of supersolubility. Following Carocca [4], we say that the subgroups H and K of a group G are mutually permutable if H commutes with every subgroup of K and K commutes with every subgroup of H. If G = HK and H and K are mutually permutable, we say that G is the mutually permutable product of the subgroups H and K. ...
Purpose
In this paper, We determine the finite group G=HKsuch that K is a supersoluble subgroup of G, and H is not a supersoluble subgroup of G.
Methods
Let p,q,rbe primes such that p<q<r, and p,qare not a divisor of r−1, and p is not a divisor of q−1. Let X be a group of order p, and let F=GF(q) and L=GF(r) such that the filed F contains a primitive pth root of unity. Let V be a simple FX-module, and let
and W also be a faithful simple LY-module. Let
,
, and
.
Results
Then, we determine that K is a supersoluble subgroup of G, and H is not a supersoluble subgroup of G.
Conclusions
We characterize the supersoluble residual of group G.
... Unfortunately, this result is not true if we replace nilpotent by supersoluble in the statement (P SL(2, 7) can be written as the product of two supersoluble subgroups). One of the main purposes of this paper is to find conditions which allow us to establish a supersoluble version of Stonehewer's Theorem. ...
... The aim of most of them has been to weaken the normality hypotheses, replacing them by permutability conditions instead. Following Carocca [7], we will say that: Every totally permutable product is clearly mutually permutable, but the converse does not hold. It is true, however, for mutually permutable products G = AB such that A ∩ B = 1 [7, Proposition 3.5]. ...
It is well known that a group which is the product of two supersoluble subgroups A and B is not supersoluble in general. Under suitable permutability conditions on A and B, we show that for any minimal normal subgroup N both AN and BN are supersoluble. We then exploit this to establish some sufficient conditions for G to be supersoluble.
... They obtained the following result: Assume that G = AB is the product of two supersoluble subgroups A and B. If every subgroup of A is permutable with every subgroup of B, then G is supersoluble (Note that a subgroup H is called permutable with a subgroup K if HK = KH). Their result was further developed and applied by many authors (see, for example, [3,4,10,12]). Recently, W. B. Guo, K. P. Shum and A. N. Skiba have introduced the concept of completely cpermutable subgroup. ...
... They obtained the following result: Assume that G = AB is the product of two supersoluble subgroups A and B. If every subgroup of A is permutable with every subgroup of B, then G is supersoluble (Note that a subgroup H is called permutable with a subgroup K if HK = KH). Their result was further developed and applied by many authors (see, for example, [3,4,10,12]). Recently, W. B. Guo, K. P. Shum and A. N. Skiba have introduced the concept of completely cpermutable subgroup. ...
Let H and T be subgroups of a finite group G. We say that H is completely c-permutable with T in G if there exists an element x ∈ H, T such that HT x = T x H. In this paper, we shall use the concept of completely c-permutable subgroups to determine the supersolubility of the product groups which are expressed by G = AB, where A and B are supersoluble subgroups of G. Some criteria of supersolubility of the above product groups are obtained and some known results are generalized.
... This shows how an approach that involves factorized groups provides a more general setting for local-global questions related to two-generated subgroups. A first extension of Thompson's theorem for products of groups was obtained by A. Carocca [5], who proved the solubility of S-connected products of soluble subgroups. This way the following general connection property turns out to be useful: [6]). ...
For a non-empty class of groups L, a finite group G=AB is said to be an L-connected product of the subgroups A and B if 〈a,b〉∈L for all a∈A and b∈B. In a previous paper, we prove that, for such a product, when L=S is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.
... Following Carocca [6], we say that G = HK is the mutually permutable product of subgroups H and K if H permutes with every subgroup of K and K permutes with every subgroup of H. The mutually permutable products of supersoluble and c-supersoluble subgroups were investigated in many works of different authors (see monograph [3]). ...
Let be a class of groups. A group G is called ca--group if its every non-abelian chief factor is simple and for every abelian chief factor H/K of G. In this paper, we investigate the structure of a finite ca--group. Properties of mutually permutable products of finite ca--groups are studied.
... Friesen [3] proved that if a group G is the product of two normal supersolvable subgroups of coprime indices, then G is also supersolvable. The search for generalisations of Baer's and Friesen's results has been a fruitful topic of investigation recently (see456). Most of the generalisations centre around replacing normality of the factors by different permutability conditions. ...
In this paper, we investigate the structure of finite groups that are products of two supersolvable groups and gain a sufficient condition for a group to be supersolvable. Our main theorem is the following: Let the group
G=HK
be the product of the subgroups
H
and
K
. Assume that
H
permutes with every maximal subgroup of
K
and
K
permutes with every maximal subgroup of
H
. If
H
is supersolvable, and
K
is nilpotent and
K
is
-permutable in
H
, where
is a complete set of Sylow subgroups of
H
, then
G
is supersolvable. Some known results are generalized.
... , G k if G i and G j are totally permutable subgroups of G, that is, every subgroup of G i permutes with every subgroup of G j for all i = j. These kinds of products have been studied extensively with a lot of properties and results available (see2345678910 and the papers cited therein). We continue the investigation about certain subgroups of pairwise mutually and pairwise totally permutable products that began by the authors in [5] and [6]. ...
We continue here our study of pairwise mutually and pairwise totally permutable products. We are looking for subgroups of the product in which the given factorization induces a factorization of the subgroup. In the case of soluble groups, it is shown that a prefactorized Carter subgroup and a prefactorized system normalizer exist. A less stringent property have
-residual,
-projector and
-normalizer for any saturated formation
including the supersoluble groups.
... In [2], Asaad and Shaalan considered products satisfying one of the following conditions. We will follow Carocca [6], and say that G = AB is the mutually permutable product of the subgroups A and B if A permutes with every subgroup of B and B permutes with every subgroup of A. We say that the product is totally permutable if every subgroup of A permutes with every subgroup of B. Essentially, the results by Asaad and Shaalan are devoted to obtaining sufficient conditions for a mutually permutable product of two supersoluble subgroups to be supersoluble. They prove in [2, Theorem 3.8] the following generalisation of Baer's theorem: ...
We investigate the structure of finite groups that are the mutually permutable product of two supersoluble groups. We show that the supersoluble residual is nilpotent and the Fitting quotient group is metabelian. These results are consequences of our main theorem, which states that such a product is supersoluble when the intersection of the two factors is core-free in the group.
... Recall that two subgroups A and B of a group G are said to permute if A B = B A, that is if A B is a subgroup of G. Following Carocca [9], we say that the group G = A B is the mutually permutable product of A and B if A permutes with every subgroup of B and B permutes with every subgroup of A. If in addition every subgroup of A permutes with every subgroup of B, then G is called the totally permutable product of A and B. Within the framework of factorized finite groups (and in the following, we consider only finite groups), these kinds of products have been widely investigated (cf.123411,12]). Since the concept of mutually permutable products is more general than that of products of normal subgroups (normal products for short), it is natural to study mutually permutable products in the context of Fitting classes (by definition, these classes of groups are closed under forming normal products and taking subnormal subgroups). ...
Let the finite group G=AB be the mutually permutable product of the subgroups A and B and let F be a Fitting class. Then the F-radicals AF and BF of the factors A and B are mutually permutable. Using this, we also prove the inclusion G′∩AFBF⩽GF, which generalizes the fact that A∈F and B∈F implies G′∈F.
... , G k if G i and G j are totally permutable subgroups of G, that is, every subgroup of G i permutes with every subgroup of G j for all i = j. These kinds of products have been studied extensively with a lot of properties and results available (see2345678910 and the papers cited therein). We continue the investigation about certain subgroups of pairwise mutually and pairwise totally permutable products that began by the authors in [5] and [6]. ...
Some results about products of pairwise mutually permutable subgroups are presented in this paper. It is shown that this kind of products behaves well with respect to some well-known classes of groups. For instance, we show that all factors have only simple chief factors if the product has this property. This is necessary but not sufficient: we need that the factors belong to the subclass of PST-groups to make sure that the product has only simple chief factors (see Theorems 5 and 6).
... This fact has been the starting point for a series of results about factorized groups in which the factors satisfy certain permutability conditions. Following Carocca [10] we say that a group G = AB is the mutually permutable product of A and B if A permutes with every subgroup of B and conversely. If, in addition, every subgroup of A permutes with every subgroup of B, we say that the group G is a totally permutable product of A and B. In a seminal paper, Asaad and Shaalan [2] study sufficient conditions for totally and mutually permutable products of two supersoluble subgroups to be supersoluble. ...
In this paper a structural theorem about mutually permutable products of finite groups is obtained. This result is used to derive some results on mutually permutable products of groups whose chief factors are simple. Some earlier results on mutually permutable products of supersoluble groups appear as particular cases.
A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson’s theorem from the perspective of factorized groups. More precisely, we study finite groups with subgroups such that is soluble for all and . In this case, the group G is said to be an -connected product of the subgroups A and B for the class of all finite soluble groups. Our Main Theorem states that is -connected if and only if [A, B] is soluble. In the course of the proof, we derive a result about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.
For a non-empty class of groups , a finite group is said to be an -connected product of the subgroups A and B if for all and . In a previous paper, we prove that for such a product, when is the class of finite soluble groups, then [A,B] is soluble. This generalizes the theorem of Thompson which states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper our result is applied to extend to finite groups previous research in the soluble universe. In particular, we characterize connected products for relevant classes of groups; among others the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Also we give local descriptions of relevant subgroups of finite groups.
In this paper we study groups G generated by two subgroups A and B such that 〈. a, b〉 is nilpotent of class at most 2 for all a∈. A and b∈. B. A detailed description of the structure of such groups is obtained, generalizing the classical result of Hopkins and Levi on 2-Engel groups.
It is proved that a finite group G=AB is supersoluble provided that A and B are such supersolube subgroups of G that every cyclic subgroup of A permutes with every Sylow subgroup of B and if in return every cyclic subgroup of B permutes with every Sylow subgroup of A.
Read More: http://www.worldscientific.com/doi/abs/10.1142/S1005386706000502?journalCode=ac
Let G be a finite group and G1, G2 are two subgroups of G. We
say that G1 and G2 are mutually permutable if G1 is permutable with every
subgroup of G2 and G2 is permutable with every subgroup of G1. We prove
that if 𝐺=𝐺1𝐺2=𝐺1𝐺3=𝐺2𝐺3 is the product of three supersolvable subgroups G1, G2, and G3, where Gi and Gj are mutually permutable for all i
and j with 𝑖≠𝑗 and the Sylow subgroups of G are abelian, then G is supersolvable. As a corollary of this result, we also prove that if G possesses three
supersolvable subgroups 𝐺𝑖(𝑖=1,2,3) whose indices are pairwise relatively
prime, and Gi and Gj are mutually permutable for all i and j with 𝑖≠𝑗, then
G is supersolvable.
The main result in the paper states the following: For a finite
group G=AB, which is the product of the soluble subgroups A
and B, if is a metanilpotent group for all
and , then the factor groups are nilpotent, F(G) denoting the Fitting subgroup of
G. A particular generalization of this result and some
consequences are also obtained. For instance, such a group G is
proved to be soluble of nilpotent length at most l+1, assuming
that the factors A and B have nilpotent length at most l. Also
for any finite soluble group G and , an element
is contained in the preimage of the hypercenter of ,
where denotes the ()th term of the Fitting series
of G, if and only if the subgroups have
nilpotent length at most k for all .
For a non-empty class of groups ℱ, two subgroups A and B of a finite group G are said to be ℱ-connected if 〈a, b〉 ∈ ℱ for all a ∈ A and b ∈ B. This paper is a study of ℱ-connection for saturated formations ℱ ⊆ (where denotes the class of all finite groups with nilpotent commutator subgroup). The class of all finite supersoluble groups constitutes an example of such a saturated formation. It is shown for example that in a finite soluble group G = AB the subgroups A and B are -connected if and only if [A, B] ⩽ F(G), where F(G) denotes the Fitting subgroup of G. Also ℱ-connected finite soluble products for any saturated formation ℱ with ℱ ⊆ are characterized.
Two subgroups X and Y of a group G are said to be conditionally permutable in G if X permutes with Yg for some element g∈G, i.e., XYg is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.
In this paper we study finite factorized groups in which certain subgroups of distinct factors permute. This often allows us to control properties of the product in terms of internal properties of the factors.
We consider the product AB of two finite mutually permutable subgroups A, B and find some subnormal subgroups of the product. This leads to local and otherwise generalized statements about products of supersolvable groups.
Saturated formations are closed under the product of subgroups which are connected by certain permutability properties.