- Research
- Open access
- Published:
On inequalities of subgroups and the structure of finite groups
Journal of Inequalities and Applications volume 2013, Article number: 427 (2013)
Abstract
Let G be a group and H be a subgroup of G. We say that H is weakly Φ-supplemented in G if G has a subgroup T such that HT = G and , where denotes the Frattini subgroup of H. In this paper, properties of this new kind of inequalities of subgroups are investigated and new characterizations of nilpotency and supersolubility of finite groups in terms of the new inequalities are obtained.
MSC:20D10, 20D15, 20D20.
1 Introduction
All groups in this paper are finite.
Let G be a group and H be a subgroup of G. H is said to be complemented in G if G has a subgroup K such that and . A lot of information about the structure of finite groups can be obtained under the assumption that some families of subgroups are complemented (cf., e.g., [1–4]). For example, a classical result of Hall is about the solubility of a group G satisfying that every Sylow subgroup of G is complemented in G [3]. A subgroup H of a group G is said to be supplemented in G if G has a subgroup T such that . It is clear that every subgroup of a group G is supplemented in G and every complemented subgroup is also a supplemented subgroup. However, supplemented subgroups may not be complemented. Based on this investigation, we introduce the following new inequalities of subgroups related closely to supplementarity of subgroups.
Definition 1.1 Let G be a group and H be a subgroup of G. H is said to be weakly Φ-supplemented in G if G has a subgroup T such that and , where is the Frattini subgroup of H.
By the definition, a complemented subgroup is still a weakly Φ-supplemented subgroup. However, the converse does not hold.
Example 1.2 Let , the quaternion group of order 8, and let H be a subgroup of G of order 4. Then H is weakly Φ-supplemented in G but not complemented in G.
This example shows that the class of all weakly Φ-supplemented subgroups is wider than the class of all complemented subgroups. Thus, a question arises naturally:
Can we characterize the structure of finite groups in terms of the weakly Φ-supplemented subgroups?
In this paper, we try to study this question and characterize the structure of groups by this new kind of inequalities of subgroups. In Section 3, we present a new criterion for the nilpotency of finite groups. In Section 4, a characterization of supersolubility of groups is given under the assumption that cyclic subgroups of order prime or 4 are weakly Φ-supplemented.
The notation and terminology in this paper are standard and the reader is referred to [5] if necessary.
2 Preliminaries
In this section, we give some lemmas which will be useful in the sequel.
Lemma 2.1 Let G be a group. Suppose that H is a weakly Φ-supplemented subgroup of G.
-
(1)
If , then H is weakly Φ-supplemented in M.
-
(2)
Suppose that and . Then is weakly Φ-supplemented in .
-
(3)
If E is a normal subgroup of G and , then is weakly Φ-supplemented in .
Proof (1) If H is weakly Φ-supplemented in G, there exists a subgroup T in G such that HT = G and . Since , and , hence H is weakly Φ-supplemented in M.
-
(2)
Since H is weakly Φ-supplemented in G, there exists a subgroup T such that HT = G and . Then it is easy to see that and . Therefore, is weakly Φ-supplemented in .
-
(3)
Assume that H is weakly Φ-supplemented in G, and let T be a subgroup of G such that
Then . Since ,
Hence, we have that
It follows that
This shows that is weakly Φ-supplemented in . □
Recall that a group G is called quasinilpotent if given any chief factor of G, every automorphism of induced by an element of G is inner; the generalized Fitting subgroup of G is the product of all normal quasinilpotent subgroups of G. The following well-known facts about the generalized Fitting subgroup of a group G will be used in our proofs (see [[6], Chapter X] and [[7], Lemma 4]).
Lemma 2.2 Let G be a group.
-
(1)
If G is quasinilpotent and N is a normal subgroup of G, then N and are quasinilpotent.
-
(2)
If N is a normal subgroup of G, then .
-
(3)
. If is soluble, then .
-
(4)
Let p be a prime and P be a normal p-subgroup of G. Then . If P is contained in , then .
3 New characterizations of nilpotency
Theorem 3.1 Let G be a group with a normal subgroup N such that is p-nilpotent. Suppose that every minimal subgroup of N of order p is contained in , and every cyclic subgroup of N with order 4 (if ) is weakly Φ-supplemented in G. Then G is p-nilpotent.
Proof Suppose that the assertion is not true, and let G be a counterexample of minimal order. Then:
-
(1)
G is a minimal non-nilpotent group and , where Q is a Sylow q-subgroup of G, is a chief factor of G and or .
Let L be a proper subgroup of G. Because and is p-nilpotent, is p-nilpotent. By the hypothesis and Lemma 2.1, every cyclic subgroup of with order 4 (if p=2) is weakly Φ-supplemented in L. Since every minimal subgroup of N of order p is contained in and , every minimal subgroup of of order p is contained in . Therefore L satisfies the hypothesis. Hence, by the choice of G, L is p-nilpotent. It follows that G is a minimal non-p-nilpotent group. Then, by [[8], Chapter IV, Theorem 5.4] and [[5], Theorem 3.4.11], G has a normal Sylow p-subgroup P satisfying that , where Q is a Sylow q-subgroup of G, is a chief factor of G and or .
-
(2)
There exists an element with order 4 of P.
It is easy to see that P is contained in N. Assume that (2) is false. Then by (1). By the hypothesis, P is contained in . Therefore G is nilpotent. This contradiction shows that (2) holds.
-
(3)
The final contradiction.
Let and . Then is weakly Φ-supplemented in G. Thus there exists a subgroup T of G such that and . Since is normal in , we have or . If , then and therefore , a contradiction. Hence and so . It follows from [[8], Chapter IV, Theorem 2.8] that G is nilpotent. This contradiction completes the proof. □
Theorem 3.2 Let G be a group with a normal subgroup N such that is nilpotent. Suppose that every minimal subgroup of is contained in and that every cyclic subgroup of with order 4 is weakly Φ-supplemented in G. Then G is nilpotent.
Proof Suppose that the statement is not true, and let G be a counterexample of minimal order.
Let M be a proper normal subgroup of G. We argue that M satisfies the hypothesis. Since , is nilpotent. By Lemma 2.2, . Hence every minimal subgroup of is contained , and every cyclic subgroup of of order 4 is weakly Φ-supplemented in M by Lemma 2.1. Therefore M satisfies the hypothesis and so it is nilpotent by the minimality of G. Furthermore, we have that is the unique maximal normal subgroup of G and is a chief factor of G. In view of Theorem 3.1, we also have , where denotes the smallest normal subgroup of G such that is nilpotent. Since [[5], Corollary 3.2.9], we have . Let , let p be the smallest prime dividing the order of F, and let P be the Sylow p-subgroup of F. Then F is a proper normal subgroup of G by Theorem 3.1 and P is normal in G. Let Q be an arbitrary Sylow q-subgroup of G with , a prime. By Lemma 2.1 and Theorem 3.1, PQ is p-nilpotent and so Q is contained in . This implies that . Hence since . Then . By Lemma 2.2, . Obviously, 2 does not divide the order of . By Lemma 2.1, fulfils the condition and so it is nilpotent by the choice of G, which shows that G is nilpotent, a finial contradiction completing the proof. □
4 New characterizations of supersolubility
Lemma 4.1 Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. Then G is p-nilpotent if and only if every cyclic subgroup of P of order prime or 4 (if P is a non-abelian 2-group) not having a supersoluble supplement in G is weakly Φ-supplemented in G.
Proof The necessity part is obvious. We prove the sufficiency. Suppose it is false. Then G is non-p-nilpotent and so G contains a minimal non-p-nilpotent subgroup A. Then A is a minimal non-nilpotent group and possesses the following properties: (1) , where is the Sylow p-subgroup of A, is the Sylow q-subgroup of A and is the smallest normal subgroup of A such that is nilpotent; (2) is a chief factor of A; (3) or 4. Without loss of generality, we suppose that . It is easy to see from Lemma 2.1 that every cyclic subgroup of A of order p or 4 (if ) not having a supersoluble supplement in A is weakly Φ-supplemented in A. Let x be an element of such that and . Then H is of order p or 4. Furthermore, one can suppose that H does not have any supersoluble supplement in A because if every cyclic subgroup of A of order p or 4 has a supersoluble supplement in A, then A is nilpotent. Then, by the hypothesis, H is weakly Φ-supplemented in G and so A has a subgroup T such that and . Since is normal in , or by (2). If the former occurs, then and , a contradiction. Suppose the latter holds, then , which implies that A is nilpotent, a final contradiction completing the proof. □
Lemma 4.2 Let P be a non-trivial normal p-subgroup of G, where p is a prime. If and every minimal subgroup of G not having a supersoluble supplement in G is weakly Φ-supplemented in G, then every chief factor of G below P is cyclic.
Proof Denote by L. Consider the factor group . We verify that is a normal subgroup of satisfying the hypothesis. Clearly, . Let be a minimal subgroup of . Then for some . Then, by the hypothesis, and so either has a supersoluble supplement in G or is weakly Φ-supplemented in G. If has a supersoluble supplement T in G, then is a supersoluble supplement of in . If is weakly Φ-supplemented in G, then G has a subgroup T such that and . Therefore
implying that is weakly Φ-supplemented in . Hence satisfies the hypothesis and consequently, by induction, every chief factor of below is cyclic provided that . Thus, every chief factor of G below P is cyclic. Now suppose that . Then P is elementary abelian of exponent p. Let N be a minimal subgroup of P. Suppose that N has a supersoluble supplement T in G. If , is supersoluble and the conclusion follows. If , then . Since P is abelian, is normal in G and so every chief factor of G below is cyclic by induction. It follows that the result holds. If N is weakly Φ-supplemented in G, then G has a subgroup T such that . As above, we have that is normal in G and every chief factor of G below is cyclic. Since , every chief factor of G below P is cyclic. Thus, the proof is complete. □
Theorem 4.3 Let G be a group. Then G is supersoluble if and only if G has a normal subgroup E such that for each non-cyclic Sylow subgroup P of E, every cyclic subgroup of P of order prime or 4 (if P is a non-abelian 2-group) without any supersoluble supplement in G is weakly Φ-supplemented in G.
Proof The necessity is clear and we only need to prove the sufficiency. We proceed the proof by induction. By Lemmas 2.1 and 4.1, E is p-nilpotent, where p is the smallest prime dividing the order of E. Let H be the Hall -subgroup of E. If H is non-trivial, then satisfies the hypothesis by Lemma 2.1 and consequently is supersoluble. By induction again, we have that G is supersoluble. Hence one can assume that E is a p-group and E is non-cyclic. Let denote the smallest normal subgroup of G such that is supersoluble. If , then G is supersoluble by Lemma 2.1 and induction. Hence we suppose that .
We assert that G is supersoluble. If not, then G is a minimal non-supersoluble group by Lemma 2.1. Therefore, G has the following properties: (1) is a non-cyclic chief factor of G; (2) or 4 (if ). If , then every chief factor of G below E is cyclic by Lemma 4.2 and so G is supersoluble, a contradiction. This shows that and . Pick such that . Set and . First suppose that L has a supersoluble supplement T in G. Then is normal in and so or . If , then , which means that E is cyclic, a contradiction. If , then is supersoluble, a contradiction. Hence L has no supersoluble supplement in G and so it is weakly Φ-supplemented in G by the hypothesis. Then there exists a subgroup T of G such that and . If , then E is cyclic as above, a contradiction. If , then and so , contradicting that . Hence G is supersoluble. □
Theorem 4.4 A group G is supersoluble if and only if G has a normal subgroup E such that is supersoluble, and for each non-cyclic Sylow subgroup P of , every cyclic subgroup of P of order prime or 4 (if P is a non-abelian 2-group) without any supersoluble supplement in G is weakly Φ-supplemented in G.
Proof The necessity is obvious and we only need to prove the sufficiency. Suppose this is not true and let G be a counterexample of with minimal. We will derive a contradiction through the following steps.
-
(1)
.
Let . Then F is soluble by Lemmas 2.1 and 4.1. Therefore by Lemma 2.2. If , then G is supersoluble by Theorem 4.3. This contradiction implies that .
-
(2)
Let p be the smallest prime dividing the order of F. Then p is odd.
Suppose that . Let P be a Sylow 2-subgroup of F, and let Q be an arbitrary Sylow q-subgroup of E, where q is an odd prime. Then P is normal in G and PQ is 2-nilpotent by the hypothesis, Lemmas 2.1 and 4.1. Hence and so . Set and . Then W is a normal subgroup of G. Since , it is immediate that every chief factor of W below P is central in W. It follows that W is quasinilpotent and so . This shows that W is nilpotent and therefore V is soluble. Thus, is nilpotent. Again, since , we see that V is nilpotent. Therefore and so . By Lemma 2.1 and the choice of G, satisfies the hypothesis and therefore is supersoluble. It follows from Theorem 4.3 that G is supersoluble, a contradiction. Hence p is an odd prime.
Now, let P be the Sylow p-subgroup of F.
-
(3)
Let D be a normal subgroup of G contained in P such that every chief factor of G below D is cyclic. Suppose that is a chief series of G below D and , where . Then .
It is easy to see that is abelian. Since , . If , then the pair satisfies the hypothesis and is less that . Thus, G is supersoluble by Lemma 2.1 and the choice of G. Hence, , as desired.
-
(4)
P is non-cyclic.
Suppose that P is cyclic. Then E stabilizes a chain of subgroups of P by (3), which implies that is a p-group. Hence . Arguing as in (2), we conclude that satisfies the hypothesis and therefore is supersoluble by the choice of G. In view of Theorem 4.3, G is supersoluble, contrary to the choice of G.
Final contradiction.
By (4), P is non-cyclic. Since p is an odd prime by (2), P contains a characteristic subgroup D of exponent p such that every non-trivial -automorphism of P induces a non-trivial automorphism of D. By Lemma 4.2, every chief factor of G below D is cyclic. Hence, by (3), is a p-group, which implies that is still a p-group by the property of D. Hence . Analogously to the discussion in (2), one can deduce that satisfies the hypothesis and consequently is supersoluble. Now, by Theorem 4.3, G is supersoluble, a final violation finishing the proof. □
References
Arad Z, Ward MB: New criteria for the solubility of finite groups. J. Algebra 1982, 77: 234–246. 10.1016/0021-8693(82)90288-5
Ballester-Bolinches A, Guo XY: On complemented subgroups of finite groups. Arch. Math. 1999, 72: 161–166. 10.1007/s000130050317
Hall P: A characteristic property of soluble groups. J. Lond. Math. Soc. 1937, 12: 188–200.
Miao L: On complemented subgroups of finite groups. Czechoslov. Math. J. 2006, 56: 1019–1028. 10.1007/s10587-006-0077-6
Guo WB: The Theory of Classes of Groups. Kluwer Academic, Dordrecht; 2000.
Huppert B, Blackburn N: Finite Groups III. Springer, Berlin; 1982.
Ballester-Bolinches A, Ezquerro LM, Skiba AN: Subgroups of finite groups with a strong cover-avoidance property. Bull. Aust. Math. Soc. 2009, 79: 499–506. 10.1017/S0004972709000100
Huppert B: Endliche Gruppen I. Springer, New York; 1967.
Acknowledgements
The authors are grateful to the referees for their helpful suggestions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11271301, 11171364, 11001226), the Scientific Research Foundation of Chongqing Municipal Education Committee (Grant No. KJ131204), the Science Fund for Creative Research Groups of Chongqing (Grant No. KJTD201321) and the Scientific Research Foundation of Chongqing University of Arts and Sciences (Grant Nos. R2012SC21, Z2012SC25).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
JL carried out the new characterizations of supersolubility. FX conceived of the study and carried out the new characterizations of nilpotency. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, J., Xie, F. On inequalities of subgroups and the structure of finite groups. J Inequal Appl 2013, 427 (2013). https://doi.org/10.1186/1029-242X-2013-427
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-427