AlgebraVolume 2013 (2013), Article ID 153691, 5 pageshttp://dx.doi.org/10.1155/2013/153691

Research Article

Finite Groups with Some -Supplemented Subgroups

Guo Zhong,¹ Liying Yang,¹ Huaquan Wei,² Xuanlong Ma,³ and Jiayi Xia¹

¹School of Mathematical Sciences, Guangxi Teachers Education University, Nanning, Guangxi 530023, China²College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China³School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received 30 March 2013; Accepted 29 May 2013

Academic Editor: Antonio M. Cegarra

Copyright © 2013 Guo Zhong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a subgroup of a finite group , a prime dividing the order of , and a Sylow -subgroup of for prime We say that is -supplemented in if there is a subgroup of such that and where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in In this paper, we characterize -nilpotency and supersolvability of under the assumption that all maximal subgroups of are -supplemented in .

1. Introduction

Throughout only finite groups are considered. Let stand for the set of all prime divisors of the order of a group . denotes the class of all supersolvable groups. char means is a characteristic subgroup of . We use conventional notions and notation, as in Robinson [1].

Let be a class of groups. is called a formation provided that (1) if and , then and (2) if and are in , then is in for all normal subgroups , of . A formation is said to be saturated if implies that .

Two subgroups and of a group are said to be permutable if . is said to be -quasinormal in if permutes with every Sylow subgroup of , that is, for any Sylow subgroup of . This concept was introduced by Kegel in [2] and has been studied widely by many authors, such as [3, 4]. An interesting question in the theory of finite groups is to determine the influence of the embedding properties of members of some distinguished families of subgroups of a group on the structure of the group. Recently, Ballester-Bolinches and Pedraza-Aguilera [5] generalized -quasinormal subgroups to -quasinormally embedded subgroups. is said to be -quasinormally embedded in provided every Sylow subgroup of is a Sylow subgroup of some -quasinormal subgroup of . By applying this concept, Ballester-Bolinches and Pedraza-Aguilera got new criteria for supersolvability of groups.

A subgroup of a group is called to be complemented in if has a subgroup such that and . is called to be supplemented in if there exists a subgroup of such that . Obviously, a complemented subgroup is a special supplemented subgroup. In recent years, it has been of interest to use supplementation properties of subgroups to characterize properties of a group. For example, the definition of a weakly -supplemented subgroup introduced by Skiba in [6]. is said to be weakly -supplemented in if has a subgroup such that and , where is the largest -quasinormal subgroup of contained in . Using this concept, many meaningful results have been obtained, such as [7–9]. More recently, the concept of -supplemented subgroups was introduced as follows.

Definition 1 (see [10, Definition 1.2]). Let be a subgroup of . We say that is -supplemented in if there exists some subgroup of such that and , where denotes the subgroup generated by all the subgroups of which are -quasinormally embedded in . We call the -core of in .

Clearly, normal subgroups, -quasinormal subgroups, -quasinormally embedded subgroups, and weakly -supplemented subgroups are all -supplemented subgroups. But the converse does not hold in general (see [11]). Based on the observation of the above concepts, we note that supplementation of some families of subgroups of a group has a strong influence on its structure.

On the other hand, the normalizer of a Sylow subgroup of a group takes an important role in studying the structure of a group. Let be a Sylow subgroup of a group ; it is natural to consider the structure of if some properties of the normalizer of are known. A classical result in this orientation is attributed to Burnside's Theorem [1, Theorem  10.1.8]. Later, Hall [12] extended it as follows: if each -element of does commute with all elements of and if also the class size of is less than , then is -nilpotent. In short, it is of significance to research into the structure of finite groups from properties of the normalizer of a Sylow subgroup of a group. In this paper, under the assumption that all maximal subgroups of a Sylow subgroup are -supplemented in , we give some new characterizations on the structure of .

2. Preliminaries

For the sake of convenience, we first cite some known results in the literature which will be useful in the following.

Lemma 2 (see [13, Lemma 2.1]). Let and be subgroups of a group . (1)If is -quasinormal in and , then is -quasinormal in .(2)Let and be -quasinormal in . Then is -quasinormal in and is -quasinormal in .(3)If is -quasinormal in , then is subnormal in .(4)If both and are -quasinormal subgroups of , then both and are -quasinormal subgroups of .

Lemma 3 (see [10, Lemma 2.6]). Suppose that is a normal subgroup of and . Then (1);(2); (3);(4)If , then .

Lemma 4 (see [10, Lemma 2.7]). Let be a subgroup of a group . (1)If is -supplemented in and , then is -supplemented in .(2)Let and . If is -supplemented in , then is -supplemented in .(3)Let be a set of primes, a -subgroup of , and a normal -subgroup of . If is -supplemented in , then is -supplemented in .

Lemma 5 (see [14]). Let be a group. (1)If is an -quasinormal -subgroup of for some prime , then .(2)Suppose that is a -subgroup of contained in . If is -quasinormally embedded in , then is -quasinormal in .

Lemma 6. Let be a subgroup of a group . If is -supplemented in and , then is weakly -supplemented in .

Proof. In fact, , where () is -quasinormally embedded in . Since , we have that is -quasinormal in by Lemma 5(2). Thus, , where is the largest -quasinormal subgroup of contained in . Consequently, is weakly -supplemented in .

Lemma 7 (see [15, Lemma 2.1]). Let be a group. If is subnormal in and is a -subgroup of , then .

From Lemma 6, the following Lemma is a direct corollary of Lemma 3.1 in [7].

Lemma 8. Suppose , where is a normal Sylow -subgroup and a Sylow -subgroup of . If and if also every maximal subgroup of is -supplemented in , then is -nilpotent.

3. Main Results

Theorem 9. Let be a group and assume is a prime dividing the order of with . If there exists a Sylow -subgroup of such that every maximal subgroup of is -supplemented in and if also is -quasinormal in , then is -nilpotent.

Proof. Suppose that the result is not true and let be a counterexample of minimal order. We will derive a contradiction in several steps. (1)  . Let , where is a prime number dividing and different from . We can see that all maximal subgroups of are -supplemented in by Lemma 4. Then meets the hypothesis of Lemma 8. It follows that is -nilpotent and thus . We know that all -elements of are contained in . If is abelian, then , which yields that is -nilpotent from Burnside's theorem [1, Theorem  10.1.8], which is a contradiction. Thus we may assume that . Since is -quasinormal in , thus is subnormal in by Lemma 2. It follows from Lemma 7 that . (2)  For any Normal Subgroup of Contained in , is -Nilpotent and is Solvable. It is clear that . For any maximal subgroup of , is a maximal subgroup of . From this hypothesis, is -supplemented in and is -quasinormal in . It follows by Lemma 4 that is -supplemented in , and is -quasinormal in by Lemma 2. As a result, meets the hypothesis of the theorem. The choice of yields that is -nilpotent. Let be a normal -complement of . If , then is a group of odd order. It follows from the Feit-Thompson theorem which asserts that every group of odd order is solvable, so is . We note that is a -group, and so is solvable. If , then is a group of odd order by . Similarly, it deduces that is solvable, too. (3) for Some Prime . Since is solvable, there exists a Sylow system of with for . By Lemmas 2 and 4, the hypothesis still holds for every . If , then and thus is -nilpotent by the choice of , whence , thereby meaning that normalizes for each . Therefore is -nilpotent and is a normal -complement of , which is a contradiction. Now, we may assume that . (4) The Final Contradiction. Let be a minimal normal subgroup of . Because is -quasinormal in and , is -quasinormal in by Lemma 2. Now we consider the quotient group . If is a -group, then , and for any maximal subgroup of , we have , where is a maximal subgroup of . From this hypothesis, is -supplemented in . Then there is a subgroup of such that and . Since , we know that As , This means that , and thus we have By Lemma 3, it follows that and thus meets the hypothesis of our theorem. The choice of implies that is -nilpotent, and so is , contradicting the fact that is a counterexample of minimal order. Consequently, we may assume that is a -group and thus . It follows by (2) that is -nilpotent. Hence we may assume that is the unique minimal normal subgroup of and . These mean that and .As is -quasinormal in , we know that by Lemma 5. Since normalizes , we get that , then since is the unique minimal normal subgroup of . It follows from (2) that is -nilpotent; thus , where . Since ,   is -nilpotent by Tate's Theorem [16, Theorem ]. Thus char which yields that ; that is, is -nilpotent, a contradiction. The final contradiction completes the proof.

Remark 10. In Theorem 9, the condition that is -quasinormal in cannot be removed.

Example 11. Let , where and . Let be a Sylow -subgroup of . By [16, II, Theorem 8.27], we know that is self-normalizing in . Clearly, every maximal subgroup of is normal in , and thus they are all -supplemented in . However, is not -nilpotent.

Remark 12. The hypothesis in Theorem 9 that is -quasinormal in still cannot be left out when is solvable and an odd prime number.

Example 13. Let be the elementary abelian -group of order . Then . It is easy to see that a subgroup of is isomorphic to . Now assume that . Let be a Sylow -subgroup of . It is clear that and thus every maximal subgroup of is -supplemented in . However, is not -nilpotent.

Theorem 14. Let be a group, a normal subgroup of such that is -nilpotent, and a Sylow -subgroup of , where is a prime dividing the order of with . If there exists a Sylow -subgroup of such that every maximal subgroup of is -supplemented in and such that is -quasinormal in , then is p-nilpotent.

Proof. Assume that the result is not true and let with subgroup be a minimal counterexample to the theorem in respect to . By Lemmas 2 and 4, we can see that every maximal subgroup of is -supplemented in and is -quasinormal in . It follows that is -nilpotent by Theorem 9. Let be the normal -complement of ; then . If , we consider the factor group with subgroup . It is clear that and . With a similar argument as in step (4) of Theorem 9, we obtain that the hypothesis still holds for with subgroup . Thereby the choice of implies that is -nilpotent. Consequently, is -nilpotent, which is a contradiction. Now we may suppose that ; that is, is a -group. Let be the normal -complement of ; this makes sense as is -nilpotent. It is easy to see that every maximal subgroup of is -supplemented in and is -quasinormal in , whence is -nilpotent by Theorem 9. As a result, char implying that is also a normal Hall -subgroup of ; that is, is -nilpotent, a contradiction. The proof of the theorem is now complete.

The following result now follows directly from Theorem 14.

Corollary 15. Let be a group and assume is a prime dividing the order of with . Suppose that is a normal subgroup of such that is -nilpotent. If there exists a Sylow -subgroup of   such that every maximal subgroup of is -quasinormally embedded or weakly -supplemented in and such that is -quasinormal in , then is -nilpotent.

The following two theorems study the supersolvability of a group.

Theorem 16. Let be a group. Suppose that for any prime dividing , there exists a Sylow -subgroup of such that every maximal subgroup of is -supplemented in and such that is -quasinormal in . Then is supersolvable.

Proof. Let be a minimal counterexample. According to Theorem 9, it is easy to see that is -nilpotent for the minimal prime dividing . Let be the normal -complement of . Then by Lemmas 2 and 4, it is clear that meets the hypothesis of the theorem, and then is supersolvable by the choice of . Let max  and . Then and . Let be a minimal normal subgroup of contained in . Thus is an elementary abelian -subgroup of . Now we consider the factor group . By Lemmas 2 and 4, we know that every maximal subgroup of is -supplemented in and is -quasinormal in . Let for any . Then for any maximal subgroup of , we know that , where is a maximal subgroup of . From the hypothesis, is -supplemented in . Then there is a subgroup of such that and . Because , we know that Since , This means that ; thus Note that by Lemma 3. Consequently, satisfies the hypothesis of the theorem. It follows that the choice of implies that is supersolvable. As the class of all supersolvable subgroups is a saturated formation, we may assume that is the unique minimal normal subgroup of contained in with , so there exists a maximal subgroup of such that and . Since ,   and normalize , whence . Thus or . If the latter case holds, then ; namely, , a contradiction. Hence and is a minimal normal subgroup of .Let be a maximal subgroup of and be a supplement of in ; then and so . This means that . However, since is normal in and is a minimal normal subgroup of , we have . Hence is the unique supplement of in . Then by Lemma 6; namely, is -quasinormal in . Thus by Lemma 5. Since , we have . It follows from the minimal normality of in that and so . Since is supersolvable, is supersolvable, a contradiction. The proof of the theorem is now complete.

Theorem 17. Let be a saturated formation containing the class of all supersolvable groups , and assume that is a group with a normal subgroup satisfying . Suppose that for any prime dividing , there exists a Sylow -subgroup of such that every maximal subgroup of is -supplemented in and such that also is -quasinormal in ; then .

Proof. Suppose that the result is not true and let with subgroup be a minimal counterexample to the theorem in respect to . By Lemmas 2 and 4, it is clear that for any prime dividing , there exists a Sylow -subgroup of such that every maximal subgroup of is -supplemented in and that is -quasinormal in . Then meets the hypothesis of Theorem 16, and thus is supersolvable. Let max  and ; then . Let be a minimal normal subgroup of contained in , and consider the quotient group . First, . With a similar argument as in the proof of Theorem 16, we can obtain that with subgroup meets the hypothesis; hence by the choice of . Consequently, we may assume that is a minimal normal subgroup of . Since constitutes a saturated formation, . Then there exists a maximal subgroup of such that and . Let be a Sylow -subgroup of . Then is a Sylow -subgroup of . Let , where is a maximal subgroup of containing . Then is a maximal subgroup of and . From this hypothesis, is -supplemented in . Let be any supplement of in ; then and . This means that . We note that is normal in and is a minimal normal subgroup of , . thus is the unique supplement of in . As a result, we know that by Lemma 6. Then is -quasinormal in . It follows that by Lemma 5. As , it is easy to see that . By the minimal normality of in , it is clear that and thus is a cyclic group of order . It follows that by [6, Lemma 2.16], a contradiction. The proof is completed.

From Theorem 17 the following corollary is immediate.

Corollary 18. Let be a saturated formation containing the class of all supersolvable groups , and let be a group with a normal subgroup satisfying . If for any prime dividing , there exists a Sylow -subgroup of such that every maximal subgroup of is -quasinormally embedded or weakly -supplemented in and such that also is -quasinormal in , then .

Acknowledgments

The authors are grateful to the referee for his or her helpful report. This research was supported by NSF of China (10961007, 11161006), NSF of Guangxi (0991101, 0991102), and 2011 higher school personnel subsidy scheme of Guangxi (5070).

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