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A characterization of supersolubility of finite groups
Journal of Inequalities and Applications volume 2012, Article number: 266 (2012)
Abstract
In this paper, we study the structure of finite groups whose minimal subgroups are λsupplemented and give a new characterization of supersolubility of finite groups.
MSC:20D10, 20D15, 20D20.
1 Introduction
All groups considered in this paper are finite.
It is an interesting topic in group theory to study the generalized permutable subgroups and generalized supplemented subgroups. The present paper is a contribution to this line of research.
Let G be a group and H be a subgroup of G. H is said to be a permutable (or quasinormal) subgroup of G if H permutes with every subgroup of G. If H permutes with every Sylow subgroup of G, then we call H an Spermutable (or Squasinormal) subgroup of G. Recently, these concepts have been generalized by many authors. In [1], Ballester and PedrazaAguilera called H an Squasinormally embedded subgroup of G if a Sylow psubgroup of H is also a Sylow psubgroup of some Squasinormal subgroup of G for each prime p dividing the order of H. In [2], Skiba introduced the concepts of weakly Spermutable subgroups and weakly Ssupplemented subgroups. H is said to be weakly Ssupplemented (weakly Spermutable) in G if G has a subgroup (subnormal subgroup) T such that G=HT and H\cap T\u2a7d{H}_{sG}, where {H}_{sG} denotes the subgroup of H generated by all those subgroups of H which are Squasinormal in G. As a generalization and unification of the above two different kinds of embedding property of subgroups, Li and Chen in [3] introduced the concept of λsupplemented subgroups. Let H be a subgroup of a group G. H is said to be λsupplemented in G if G has a subgroup T such that G=HT and H\cap T\u2a7d{H}_{SE}. In this definition, {H}_{SE} denotes the subgroup of H generated by all those subgroups of H which are Squasinormally embedded in G. Using this idea, some new characterizations of psupersolubility of finite groups are obtained in [3] under the assumption that the maximal subgroups of the Sylow subgroups of some special subgroups are λsupplemented.
In this paper, we study the structure of groups with some λsupplemented minimal subgroups and give a new characterization of supersolubility of finite groups. In fact, the main result established in this paper provides a new criterion, in terms of the λsupplemented cyclic subgroups of order prime or 4 contained in the generalized Fitting subgroups, for a group to be contained in a saturated formation containing all supersoluble groups.
We prove the main result in Section 3. Section 4 includes some applications of our main result, which unify and extend many previous known results; and the paper concludes with a partial answer to a question of Skiba.
The notation and terminology in this paper are standard. The reader is referred to [4] if necessary.
2 Preliminaries
We cite here some known results which are useful in the sequel.
Lemma 2.1 [[3], Lemma 2.3]
Let G be a group and H\u2a7dK\u2a7dG.

(1)
Suppose that H is normal in G. Then K/H is λsupplemented in G/H if and only if K is λsupplemented in G.

(2)
If H is λsupplemented in G, then H is λsupplemented in K.

(3)
Suppose that H is normal in G. Then NH/H is λsupplemented in G/H for every λsupplemented subgroup N of G satisfying (N,H)=1.
Lemma 2.2 [[3], Lemma 2.2]
Suppose that U is Squasinormally embedded in a group G. If U\u2a7d{O}_{p}(G), then U is Squasinormal in G.
Lemma 2.3 [[5], Lemma 2.8]
Let H be a normal subgroup of a group G. Then H\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G) if and only if H/\mathrm{\Phi}(H)\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G).
Lemma 2.4 [[6], Theorem 3.1]
Let p be the smallest prime dividing the order of a group G and P be a Sylow psubgroup of G. Then G is pnilpotent if and only if every cyclic subgroup H of P of order p or order 4 (if P is a nonAbelian 2group and H\u2288{Z}_{\mathrm{\infty}}(G)) is λsupplemented in G provided that H does not possess any supersoluble supplement in G.
Lemma 2.5 [[6], Theorem 3.2]
Let G be a group with a normal subgroup E such that G/E is supersoluble. Suppose that for every noncyclic Sylow subgroup P of E, every cyclic subgroup H of P of prime order or order 4 (if P is a nonAbelian 2group and H\u2288{Z}_{\mathrm{\infty}}(G)) is λsupplemented in G provided that H does not possess any supersoluble supplement in G. Then G is supersoluble.
Lemma 2.6 [[7], Chapter X]
Let G be a group. Then

(1)
If G is quasinilpotent and N is a normal subgroup of G, then N and G/N are quasinilpotent.

(2)
If N is a normal subgroup of G, then {F}^{\ast}(N)=N\cap {F}^{\ast}(G).

(3)
F(G)\u2a7d{F}^{\ast}(G)={F}^{\ast}({F}^{\ast}(G)). Moreover, if {F}^{\ast}(G) is soluble, then {F}^{\ast}(G)=F(G).
3 Main results
Lemma 3.1 Let N be a nontrivial normal psubgroup of a group G of exponent p. Suppose that every minimal subgroup of N without any supersoluble supplement in G is λsupplemented in G. Then N\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G).
Proof Assume that \mathrm{\Phi}(N)\ne 1. We check that N/\mathrm{\Phi}(N) satisfies the hypothesis. Let H/\mathrm{\Phi}(N) be a minimal subgroup of N/\mathrm{\Phi}(N). Then H/\mathrm{\Phi}(N)=\u3008x\u3009\mathrm{\Phi}(N)/\mathrm{\Phi}(N), where x\in H\setminus \mathrm{\Phi}(N). By the hypothesis, x is of order p. If \u3008x\u3009 has a supersoluble supplement T in G, then T\mathrm{\Phi}(N)/\mathrm{\Phi}(N) is a supersoluble supplement of H/\mathrm{\Phi}(N) in G/\mathrm{\Phi}(N). Suppose that \u3008x\u3009 does not have a supersoluble supplement in G. Then \u3008x\u3009 is λsupplemented in G by the hypothesis and so G has a subgroup T such that G=\u3008x\u3009T and \u3008x\u3009\cap T\u2a7d{\u3008x\u3009}_{SE}. If {\u3008x\u3009}_{SE}=1, then \u3008x\u3009\mathrm{\Phi}(N)/\mathrm{\Phi}(N)\cap T\mathrm{\Phi}(N/\mathrm{\Phi}(N))=1. If {\u3008x\u3009}_{SE}\ne 1, then \u3008x\u3009 is Squasinormal in G by Lemma 2.2 and therefore H/\mathrm{\Phi}(N) is Squasinormal in G/\mathrm{\Phi}(N). Thus, we have that H/\mathrm{\Phi}(N) is λsupplemented in G/\mathrm{\Phi}(N). Hence, N/\mathrm{\Phi}(N) satisfies the hypothesis and consequently N/\mathrm{\Phi}(N)\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G/\mathrm{\Phi}(N)) by induction. By Lemma 2.3, N\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G). Now we consider the case \mathrm{\Phi}(N)=1. Let H/K be any chief factor of G with H\u2a7dN and L/K be a minimal subgroup of H/K which is normal in some Sylow psubgroup of G/K. Then L=\u3008x\u3009K for some x\in L\setminus K. Suppose that \u3008x\u3009 has a supersoluble supplement T in G. Then \u3008x\u3009\u2a7dT or \u3008x\u3009\cap T=1. If \u3008x\u3009\u2a7dT, then H/K is of order p since G/K=T/K is a supersoluble group. Assume that \u3008x\u3009\cap T=1. Then N\cap T is normal in G and N/(N\cap T) is of order p. By Lemma 2.1, N\cap T satisfies the hypothesis and so N\cap T\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G) by induction. It follows that N\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G) since N/(N\cap T) is of order p. If \u3008x\u3009 does not have a supersoluble supplement in G, then by the hypothesis, G has a subgroup T such that G=\u3008x\u3009T and \u3008x\u3009\cap T\u2a7d{\u3008x\u3009}_{SE}. If \u3008x\u3009\cap T=1, then N\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G) as above. If {\u3008x\u3009}_{SE}\ne 1, then \u3008x\u3009 is Squasinormal in G by Lemma 2.2. Hence, {O}^{p}(G)\subseteq {N}_{G}(\u3008x\u3009). Since L/K=\u3008x\u3009K/K is also a normal subgroup of some Sylow psubgroup of G/K, we obtain that L/K is normal in G/K and so H/K is of order p, which implies that N\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G). Thus, the proof is complete. □
Theorem 3.2 A group G is supersoluble if and only if G has a normal subgroup E such that G/E is supersoluble and for every noncyclic Sylow subgroup P of {F}^{\ast}(E), every cyclic subgroup H of P of prime order or order 4 (if P is a nonAbelian 2group and H\u2288{Z}_{\mathrm{\infty}}(G)) is λsupplemented in G provided that H does not possess any supersoluble supplement in G.
Proof The necessity is obvious and we consider only the sufficiency. Suppose it is false and let G be a counterexample with G+E minimal. Then G has the following properties.

(1)
F={F}^{\ast}(E)=F(E)\ne E.
By Lemma 2.4, {F}^{\ast}(E) is soluble. Hence, {F}^{\ast}(E)=F(E) by Lemma 2.6. If {F}^{\ast}(E)=E, then by Lemma 2.5, we conclude that G is supersoluble, a contradiction. Therefore, {F}^{\ast}(E) is a proper subgroup of E.

(2)
Let p be the smallest prime dividing the order of F and P be a Sylow psubgroup of F. Then p>2.
Assume that p=2. Write V/P={F}^{\ast}(E/P). Let Q be a Sylow qsubgroup of E, where q\ne 2. Then PQ is 2nilpotent by the hypothesis and Lemma 2.4. Since P is normal in G, PQ is nilpotent. It is immediate that {O}^{2}(E)\u2a7d{C}_{E}(P). Consider the normal subgroup W={O}^{2}(V)P of E. Then W/P is a quasinilpotent group by Lemma 2.6. We claim that every chief factor of W below P is central in W. In fact, let H/K be a chief factor of W with H\u2a7dP and {W}_{p}/P a Sylow psubgroup of W/P. Then {W}_{p} is a Sylow psubgroup of W and H/K\cap Z({W}_{p}/K)\ne 1. Let L/K be a subgroup of order p contained in H/K\cap Z({W}_{p}/K). Then L/K is centralized by {W}_{p}/K. Besides, since {O}^{2}(E)\le {C}_{E}(P), L/K is centralized by {W}_{q}K/K for every Sylow qsubgroup {W}_{q} of W with q\ne p. Hence, L/K is centralized by W/K. It follows that H/K=L/K is a central chief factor of W, as desired. Therefore W is a quasinilpotent group and W\u2a7d{F}^{\ast}(E)=F(E). Thus, W is nilpotent and V/F(E) is a 2group, which shows that V is soluble. Hence, V/P={F}^{\ast}(E/P) is nilpotent. Let R be a Sylow 2subgroup of V. Then R is normal in V. Let H be a Hall 2^{′}subgroup of V. Then H stabilizes the series R\u2a7eP\u2a7e1 and so H\u2a7d{C}_{V}(R) (see [[4], Lemma 3.2.3] or [[8], Ch.5, Theorem 3.2]). Now we have that V is nilpotent and so V=F. By Lemma 2.1, G/P satisfies the hypothesis. The choice of G implies that G/P is supersoluble and so G is supersoluble by Lemma 2.5, a contradiction. Hence, p is an odd prime.

(3)
If for some normal subgroup D of G we have D\u2a7dP\cap {Z}_{\mathrm{\infty}}^{\mathcal{U}}(G), 1={D}_{0}\u2a7d{D}_{1}\u2a7d\cdots \u2a7d{D}_{t}=D is a chief series of G below D and C={C}_{1}\cap {C}_{2}\cap \cdots \cap {C}_{t}, where {C}_{i}={C}_{G}({D}_{i}/{D}_{i1}), then E\le C.
Since D\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G), G/{C}_{G}({D}_{i}/{D}_{i1}) is cyclic and so G/C is Abelian. Thus, G/(E\cap C) is supersoluble by the hypothesis. By (1) {F}^{\ast}(E)=F(E)\le E\cap C, from which we conclude that {F}^{\ast}(E\cap C)={F}^{\ast}(E) by Lemma 2.6. Hence, the hypothesis is still true for (G,E\cap C). The minimality of G+E implies that E\le C.

(4)
P is not cyclic.
If not, then E stabilizes a chain of subgroups of P by (3). It follows from [[4], Lemma 3.2.3] that E/{C}_{E}(P) is a pgroup. Thus, {O}^{p}(E)\le {C}_{E}(P). Similar to (2), we have that the pair (G/P,E/P) satisfies the hypothesis. Therefore, the minimal choice of (G,E) implies that G/P is supersoluble. Hence, G is supersoluble by Lemma 2.5, a contradiction.
Final contradiction.
By (4) and [[8], Ch.5, Theorem 3.13], P possesses a characteristic subgroup D of exponent p such that every nontrivial {p}^{\prime}automorphism of P induces a nontrivial automorphism of D. Then D\u2a7d{Z}_{\mathrm{\infty}}^{\mathcal{U}}(G) by Lemma 3.1. Let 1={D}_{0}\u2a7d{D}_{1}\u2a7d\cdots \u2a7d{D}_{t}=D be a chief series of G below D and C={C}_{1}\cap {C}_{2}\cap \cdots \cap {C}_{t} where {C}_{i}={C}_{G}({D}_{i}/{D}_{i1}). Then E\u2a7dC via (3). Thus, E/{C}_{E}(D) is a pgroup, which implies that E/{C}_{E}(P) is also a pgroup. Equivalently, we have {O}^{p}(E)\u2a7d{C}_{E}(P). Using an analogous argument as in (2), we have that G/P is supersoluble and consequently G is supersoluble by Lemma 2.5, which violates the choice of G. □
4 Some applications
Since many relevant families of subgroups, such as normal subgroups, (S)quasinormal subgroups (or (S)permutable subgroups), cnormal subgroups, complemented subgroups, csupplemented subgroups, weakly Ssupplemented subgroups and Squasinormally embedded subgroups, enjoy the λsupplementary property, a lot of nice results follow from Theorem 3.2.
Recall first some concepts of subgroups mentioned above. Let H be a subgroup of a group G. We call H a complemented subgroup of G if there exists a subgroup T of G such that G=HT and the intersection of H and T is trivial. H is said to be cnormal (csupplemented) in G if G has a normal subgroup (a subgroup) T such that G=HT and H\cap T\u2a7d{H}_{G}, where {H}_{G} denotes the largest normal subgroup of G contained in H (see [9–11]).
Now, we give three corollaries which follow directly from Theorem 3.2.
Corollary 4.1 [12]
If every minimal subgroup of a group G of odd order is normal in G, then G is supersoluble.
Corollary 4.2 [13]
Let G be a group with a normal soluble subgroup E such that G/E is supersoluble. If all subgroups of F(E) of prime order or order 4 are cnormal in G, then G is supersoluble.
Corollary 4.3 [14]
Let G be a soluble group and E be a normal subgroup of G such that G/E is supersoluble. If every minimal subgroup of F(E) is complemented in G, then G is supersoluble.
The following result can be established by applying a similar argument as that in the proof of Theorem 3.2.
Theorem 4.4 Let ℱ be a saturated formation containing all supersoluble groups. A group G\in \mathcal{F} if and only if there exists a normal subgroup E of G such that G/E\in \mathcal{F} and for every noncyclic Sylow subgroup P of {F}^{\ast}(E), every cyclic subgroup H of P of prime order or order 4 (if P is a nonAbelian 2group and H\u2288{Z}_{\mathrm{\infty}}(G)) is λsupplemented in G provided that H does not possess any supersoluble supplement in G.
By Theorem 4.4, we have
Corollary 4.5 [15]
Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal soluble subgroup E such that G/E\in \mathcal{F}. If all minimal subgroups and all cyclic subgroups with order 4 of F(E) are cnormal in G, then G\in \mathcal{F}.
Corollary 4.6 [16]
Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal subgroup E such that G/E\in \mathcal{F}. If all minimal subgroups and all cyclic subgroups with order 4 of {F}^{\ast}(E) are cnormal in G, then G\in \mathcal{F}.
Corollary 4.7 [17]
Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal subgroup E such that G/E\in \mathcal{F}. If every minimal subgroup and each cyclic subgroup with order 4 of {F}^{\ast}(E) is csupplemented in G, then G\in \mathcal{F}.
Corollary 4.8 [18]
Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal soluble subgroup E such that G/E\in \mathcal{F}. If all minimal subgroups of F(E\cap {G}^{\prime}) are complemented in G, then G\in \mathcal{F}.
Corollary 4.9 [19]
Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal soluble subgroup E such that G/E\in \mathcal{F}. If all minimal subgroups and all cyclic subgroups with order 4 of F(E) are Spermutable in G, then G\in \mathcal{F}.
Corollary 4.10 [20]
Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal subgroup E such that G/E\in \mathcal{F}. If every cyclic subgroup of {F}^{\ast}(E) of prime order or order 4 is Spermutable in G, then G\in \mathcal{F}.
Corollary 4.11 [21]
Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal subgroup E such that G/E\in \mathcal{F}. If each minimal subgroup of {F}^{\ast}(E) of order prime or 4 is Squasinormally embedded in G, then G\in \mathcal{F}.
In [2], Skiba proposed the following question.
Question 4.12 Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal subgroup E such that G/E\in \mathcal{F}. Suppose that every noncyclic Sylow subgroup P of {F}^{\ast}(E) has a subgroup D such that 1<D<P and all subgroups H of P with order H=D and with order 2D (if P is a nonAbelian 2subgroup and P:D>2) are weakly ssupplemented in G. Is then G\in \mathcal{F}?
Our last corollary gives a positive answer to the above question in the case of minimal subgroups.
Corollary 4.13 Let ℱ be a saturated formation containing all supersoluble groups and G be a group with a normal subgroup E such that G/E\in \mathcal{F}. If each cyclic subgroup of {F}^{\ast}(E) of order prime or 4 is weakly Ssupplemented in G, then G\in \mathcal{F}.
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Acknowledgements
The author would like to express his sincere thanks to the referees whose careful reading and important comments on this article led to a number of improvements. This work was supported by the Scientific Research Foundation of Chongqing University of Arts and Sciences (Grant Nos. R2012SC21, Z2012SC25) and the National Natural Science Foundation of China (Grant Nos. 11271301, 11171364, 11001226).
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Li, J. A characterization of supersolubility of finite groups. J Inequal Appl 2012, 266 (2012). https://doi.org/10.1186/1029242X2012266
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DOI: https://doi.org/10.1186/1029242X2012266