# Finite Groups with SomeSE-Supplemented Subgroups

Authors: Guo Zhong; Liying Yang; Huaquan Wei; Xuanlong Ma; Jiayi Xia
Journal: Algebra
 (2013)

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## Abstract

LetH be a subgroup of a finite group G, p a prime dividing the order of G, and P a Sylow p-subgroup of G for prime p. We say that H is SE-supplemented in G if there is a subgroup K of G such that G=HK and H∩K≤HseG, where HseG denotes the subgroup of H generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, we characterize p-nilpotency and supersolvability of G under the assumption that all maximal subgroups of P are SE-supplemented in NG(P).


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## Body

## 1. Introduction

Throughout only finite groups are considered. Letπ(G) stand for the set of all prime divisors of the order of a group G. 𝒰 denotes the class of all supersolvable groups. H char G means H is a characteristic subgroup of G. We use conventional notions and notation, as in Robinson [1].Letℱ be a class of groups. ℱ is called a formation provided that (1) if G∈ℱ and H⊲G, then G/H∈ℱ and (2) if G/M and G/N are in ℱ, then G/M∩N is in ℱ for all normal subgroups M, N of G. A formation ℱ is said to be saturated if G/Φ(G)∈ℱ implies that G∈ℱ.Two subgroupsH and K of a group G are said to be permutable if HK=KH. H is said to be S-quasinormal in G if H permutes with every Sylow subgroup of G, that is, HP=PH for any Sylow subgroup P of G. 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 S-quasinormal subgroups to S-quasinormally embedded subgroups. H is said to be S-quasinormally embedded in G provided every Sylow subgroup of H is a Sylow subgroup of some S-quasinormal subgroup of G. By applying this concept, Ballester-Bolinches and Pedraza-Aguilera got new criteria for supersolvability of groups.A subgroupH of a group G is called to be complemented in G if G has a subgroup K such that G=HK and H∩K=1. H is called to be supplemented in G if there exists a subgroup T of G such that G=HT. 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 S-supplemented subgroup introduced by Skiba in [6]. H is said to be weakly SE-supplemented in G if G has a subgroup U such that HU=G and H∩U≤HsG, where HsG is the largest S-quasinormal subgroup of G contained in H. Using this concept, many meaningful results have been obtained, such as [7–9]. More recently, the concept of SE-supplemented subgroups was introduced as follows.Definition 1 (see [10, Definition 1.2]).
LetH be a subgroup of G. We say that H is SE-supplemented in G if there exists some subgroup T of G such that HT=G and H∩T≤HseG, where HseG denotes the subgroup generated by all the subgroups of H which are S-quasinormally embedded in G. We call HseG the SE-core of H in G.Clearly, normal subgroups,S-quasinormal subgroups, S-quasinormally embedded subgroups, and weakly S-supplemented subgroups are all SE-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. LetP be a Sylow subgroup of a group G; it is natural to consider the structure of G if some properties of the normalizer NG(P) of P 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 p′-element of NG(P) does commute with all elements of P and if also the class size of P is less than p, then G is p-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 P are SE-supplemented in NG(P), we give some new characterizations on the structure of G.


## 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]).
LetH and K be subgroups of a group G.(1)
IfH is S-quasinormal in G and H≤M≤G, then H is S-quasinormal in M.(2)
LetN⊲G and H be S-quasinormal in G. Then HN is S-quasinormal in G and HN/N is S-quasinormal in G/N.(3)
IfH is S-quasinormal in G, then H is subnormal in G.(4)
If bothH and K are S-quasinormal subgroups of G, then both H∩K and 〈H,K〉 are S-quasinormal subgroups of G.Lemma 3 (see [10, Lemma 2.6]).
Suppose thatN is a normal subgroup of G and H≤K≤G. Then(1)
HseG⊴H;(2)
HseG≤HseK;(3)
HseGN/N≤(HN/N)se(G/N);(4)
If(|N|,|H|)=1, then HseGN/N=(HN/N)se(G/N).Lemma 4 (see [10, Lemma 2.7]).
LetH be a subgroup of a group G.(1)
IfH is SE-supplemented in G and H≤M≤G, then H is SE-supplemented in M.(2)
LetN⊲G and N≤H. If H is SE-supplemented in G, then H/N is SE-supplemented in G/N.(3)
Letπ be a set of primes, H a π-subgroup of G, and N a normal π′-subgroup of G. If H is SE-supplemented in G, then HN/N is SE-supplemented in G/N.Lemma 5 (see [14]).
LetG be a group.(1)
IfP is an S-quasinormal p-subgroup of G for some prime p, then NG(P)≥Op(G).(2)
Suppose thatH is a p-subgroup of G contained in Op(G). If H is S-quasinormally embedded in G, then H is S-quasinormal in G.Lemma 6.
LetH be a subgroup of a group G. If H is SE-supplemented in G and H≤Op(G), then H is weakly SE-supplemented in G.Proof.
In fact,HseG=〈H1,H2,…,Hs〉, where Hi (i=1,2,…,s) is S-quasinormally embedded in G. Since H≤Op(G), we have that Hi is S-quasinormal in G by Lemma 5(2). Thus, HseG≤HsG, where HsG is the largest S-quasinormal subgroup of G contained in H. Consequently, H is weakly SE-supplemented in G.Lemma 7 (see [15, Lemma 2.1]).
LetG be a group. If A is subnormal in G and A is a π-subgroup of G, then A≤Oπ(G).From Lemma6, the following Lemma is a direct corollary of Lemma 3.1 in [7].Lemma 8.
SupposeG=PQ, where P is a normal Sylow p-subgroup and Q a Sylow q-subgroup of G. If (|G|,p-1)=1 and if also every maximal subgroup of P is SE-supplemented in G, then G is p-nilpotent.


## 3. Main Results

Theorem 9.
LetG be a group and assume p is a prime dividing the order of G with (|G|,p-1)=1. If there exists a Sylow p-subgroup P of G such that every maximal subgroup of P is SE-supplemented in NG(P) and if also P′ is S-quasinormal in G, then G is p-nilpotent.Proof.
Suppose that the result is not true and letG be a counterexample of minimal order. We will derive a contradiction in several steps.
(1)Op(G)≥P′≠1. Let Q∈Sylq(NG(P)), where q is a prime number dividing |NG(P)| and different from p. We can see that all maximal subgroups of P are SE-supplemented in PQ by Lemma 4. Then PQ meets the hypothesis of Lemma 8. It follows that PQ is p-nilpotent and thus Q≤CG(P). We know that all p′-elements of NG(P) are contained in CG(P). If P is abelian, then NG(P)=CG(P), which yields that G is p-nilpotent from Burnside's theorem [1, Theorem  10.1.8], which is a contradiction. Thus we may assume that P′≠1. Since P′ is S-quasinormal in G, thus P′ is subnormal in G by Lemma 2. It follows from Lemma 7 that 1≠P′≤Op(G).
(2)For any Normal SubgroupNofGContained inP, G/Nisp-Nilpotent andGis Solvable. It is clear that (|G/N|,p-1)=1. For any maximal subgroup P1/N of P/N, P1 is a maximal subgroup of P. From this hypothesis, P1 is SE-supplemented in NG(P) and P′ is S-quasinormal in G. It follows by Lemma 4 that P1/N is SE-supplemented in NG(P)/N=NG/N(P/N), and (P/N)′=P′N/N is S-quasinormal in G/N by Lemma 2. As a result, G/N meets the hypothesis of the theorem. The choice of G yields that G/N is p-nilpotent. Let M/N be a normal p-complement of G/N. If p=2, then M/N 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 M. We note that G/M is a 2-group, and so G is solvable. If p≠2, then G is a group of odd order by (|G|,p-1)=1. Similarly, it deduces that G is solvable, too.
(3)|G|=paqbfor Some Primeq≠p. Since G is solvable, there exists a Sylow system P1={P,P2,…,Ps} of G with Gi=PPi for 2≤i≤s. By Lemmas 2 and 4, the hypothesis still holds for every Gi. If |π(G)|>2, then Gi<G and thus Gi is p-nilpotent by the choice of G, whence Pi⊴Gi, thereby meaning that P normalizes Pi for each 2≤i≤s. Therefore G is p-nilpotent and K=P2P3⋯Ps is a normal p-complement of G, which is a contradiction. Now, we may assume that |G|=paqb.
(4)The Final Contradiction. Let N be a minimal normal subgroup of G. Because P′ is S-quasinormal in G and N⊴G, P′N/N is S-quasinormal in G/N by Lemma 2. Now we consider the quotient group G/N. If N is a q-group, then PN/N∈Sylp(G/N), and for any maximal subgroup M/N of PN/N, we have M=P1N, where P1 is a maximal subgroup of P. From this hypothesis, P1 is SE-supplemented in NG(P). Then there is a subgroup T of G such that NG(P)=P1T and P1∩T≤(P1)seNG(P). Since P∈Sylp(G), we know that
(1)NG/N(PNN)=NG(P)NN=(P1NN)(TNN).
As (|P1|,|N|)=1,
(2)|P1∩TN|=(|P1|·|TN|p)|P1TN|p=(|P1|·|T|p)|NG(P)N|p=(|P1|·|T|p)|NG(P)|p=|P1∩T|.
This means that P1∩TN=P1∩T, and thus we have
(3)(P1NN)∩(TNN)=(P1N∩TN)N=(P1∩TN)NN=(P1∩T)NN≤(P1)seNG(P)NN.
By Lemma 3, it follows that (P1)seNG(P)N/N≤(P1N/N)se(NG/N(PN/N)) and thus G/N meets the hypothesis of our theorem. The choice of G implies that G/N is p-nilpotent, and so is G, contradicting the fact that G is a counterexample of minimal order. Consequently, we may assume that N is a p-group and thus N≤P. It follows by (2) that G/N is p-nilpotent. Hence we may assume that N is the unique minimal normal subgroup of G and N≰Φ(G). These mean that Φ(G)=1 and Op(G)=F(G)=N.
AsP′ is S-quasinormal in G, we know that NG(P′)≥Op(G) by Lemma 5. Since P normalizes P′, we get that P′⊴G, then P′=Op(G)=N since N is the unique minimal normal subgroup of G. It follows from (2) that G/Op(G) is p-nilpotent; thus Op(G)Q⊴G, where Q∈Sylq(G). Since Op(G)Q∩P=Op(G)=P′≤Φ(P),  Op(G)Q is p-nilpotent by Tate's Theorem [16, Theorem 4.4.7]. Thus Q char Op(G)Q⊴G which yields that Q⊴G; that is, G is p-nilpotent, a contradiction. The final contradiction completes the proof.Remark 10.
In Theorem9, the condition that P′ is S-quasinormal in G cannot be removed.Example 11.
LetG=PSL2(q), where q>1 and q≡±1(mod8). Let P be a Sylow 2-subgroup of G. By [16, II, Theorem 8.27], we know that P is self-normalizing in PSL2(q). Clearly, every maximal subgroup of P is normal in NG(P)=P, and thus they are all SE-supplemented in NG(P). However, G is not 2-nilpotent.Remark 12.
The hypothesis in Theorem9 that P′ is S-quasinormal in G still cannot be left out when G is solvable and p an odd prime number.Example 13.
LetH be the elementary abelian 3-group of order 27. Then H=Z3×Z3×Z3. It is easy to see that a subgroup of Aut(H) is isomorphic to Z13⋊Z3. Now assume that G=(Z3×Z3×Z3)⋊(Z13⋊Z3). Let P3 be a Sylow 3-subgroup of G. It is clear that NG(P3)=P3 and thus every maximal subgroup of P3 is SE-supplemented in NG(P3)=P3. However, G is not 3-nilpotent.Theorem 14.
LetG be a group, H a normal subgroup of G such that G/H is p-nilpotent, and P a Sylow p-subgroup of H, where p is a prime dividing the order of G with (|G|,p-1)=1. If there exists a Sylow p-subgroup P of H such that every maximal subgroup of P is SE-supplemented in NG(P) and such that P′ is S-quasinormal in G, then G is p-nilpotent.Proof.
Assume that the result is not true and letG with subgroup H be a minimal counterexample to the theorem in respect to |G|+|H|. By Lemmas 2 and 4, we can see that every maximal subgroup of P is SE-supplemented in NH(P) and P′ is S-quasinormal in H. It follows that H is p-nilpotent by Theorem 9. Let M be the normal p-complement of H; then M⊴G. If M≠1, we consider the factor group G/M with subgroup H/M. It is clear that H=PM and (|P|,|M|)=1. With a similar argument as in step (4) of Theorem 9, we obtain that the hypothesis still holds for G/M with subgroup H/M. Thereby the choice of G implies that G/M is p-nilpotent. Consequently, G is p-nilpotent, which is a contradiction. Now we may suppose that M=1; that is, H=P is a p-group. Let T/P be the normal p-complement of G/P; this makes sense as G/P=G/H is p-nilpotent. It is easy to see that every maximal subgroup of P is SE-supplemented in NT(P) and P′ is S-quasinormal in T, whence T is p-nilpotent by Theorem 9. As a result, Tp′ char T⊴G implying that Tp′ is also a normal Hall p′-subgroup of G; that is, G is p-nilpotent, a contradiction. The proof of the theorem is now complete.The following result now follows directly from Theorem14.Corollary 15.
LetG be a group and assume p is a prime dividing the order of G with (|G|,p-1)=1. Suppose that H is a normal subgroup of G such that G/H is p-nilpotent. If there exists a Sylow p-subgroup P of  H such that every maximal subgroup of P is S-quasinormally embedded or weakly SE-supplemented in NG(P) and such that P′ is S-quasinormal in G, then G is p-nilpotent.The following two theorems study the supersolvability of a group.Theorem 16.
LetG be a group. Suppose that for any prime p dividing |G|, there exists a Sylow p-subgroup P of G such that every maximal subgroup of P is SE-supplemented in NG(P) and such that P′ is S-quasinormal in G. Then G is supersolvable.Proof.
LetG be a minimal counterexample. According to Theorem 9, it is easy to see that G is p-nilpotent for the minimal prime p dividing |G|. Let K be the normal p-complement of G. Then by Lemmas 2 and 4, it is clear that K meets the hypothesis of the theorem, and then K is supersolvable by the choice of G. Let q= max π(K) and Q∈Sylq(K). Then Q∈Sylq(G) and Q⊴G. Let N be a minimal normal subgroup of G contained in Q. Thus N is an elementary abelian q-subgroup of G. Now we consider the factor group G/N. By Lemmas 2 and 4, we know that every maximal subgroup Q1/N of Q/N is SE-supplemented in NG/N(Q/N)=NG(Q)/N=G/N and (Q/N)′=Q′N/N is S-quasinormal in G/N. Let RN/N∈Sylr(G/N) for any r≠q. Then for any maximal subgroup T/N of RN/N, we know that T=R1N, where R1 is a maximal subgroup of R. From the hypothesis, R1 is SE-supplemented in NG(R). Then there is a subgroup K of NG(R) such that NG(R)=R1K and R1∩K≤(R1)seNG(R). Because R∈Sylr(G), we know that
(4)NG/N(RNN)=NG(R)NN=(R1NN)(KNN).
Since (|R1|,|N|)=1,
(5)|R1∩KN|=(|R1|·|KN|r)|R1KN|r=(|R1|·|K|r)|NG(R)N|r=(|R1|·|K|r)|NG(R)|r=|R1∩K|.
This means that R1∩KN=R1∩K; thus
(6)(R1NN)∩(KNN)=(R1N∩KN)N=(R1∩KN)NN=(R1∩K)NN≤(R1)seNG(R)NN.
Note that (R1)seNG(R)N/N=(R1N/N)se(NG/N(RN/N)) by Lemma 3. Consequently, G/N satisfies the hypothesis of the theorem. It follows that the choice of G implies that G/N is supersolvable. As the class of all supersolvable subgroups is a saturated formation, we may assume that N is the unique minimal normal subgroup of G contained in Q with N≰Φ(G), so there exists a maximal subgroup M of G such that G=MN and M∩N=1. Since Q=Oq(G)≤F(G)≤CG(N),  N and M normalize Q∩M, whence Q∩M⊴G. Thus Q∩M=1 or N≤Q∩M. If the latter case holds, then N≤M; namely, G=NM=M, a contradiction. Hence Q∩M=1 and Q=N is a minimal normal subgroup of G.
LetQ1 be a maximal subgroup of Q and T be a supplement of Q1 in G; then Q1T=G and so Q=Q∩Q1T=Q1(Q∩T). This means that Q∩T=1. However, since Q∩T is normal in G and Q is a minimal normal subgroup of G, we have Q∩T=Q. Hence T=G is the unique supplement of Q1 in G. Then Q1∩T≤(Q1)sG by Lemma 6; namely, (Q1)sG=Q1 is S-quasinormal in G. Thus NG(Q1)≥Oq(G) by Lemma 5. Since Q1⊲Q, we have Q1⊲QOq(G)=G. It follows from the minimal normality of Q in G that Q1=1 and so |Q|=q. Since G/Q is supersolvable, G is supersolvable, a contradiction. The proof of the theorem is now complete.Theorem 17.
Letℱ be a saturated formation containing the class of all supersolvable groups 𝒰, and assume that G is a group with a normal subgroup H satisfying G/H∈ℱ. Suppose that for any prime p dividing |H|, there exists a Sylow p-subgroup P of H such that every maximal subgroup of P is SE-supplemented in NG(P) and such that also P′ is S-quasinormal in G; then G∈ℱ.Proof.
Suppose that the result is not true and letG with subgroup H be a minimal counterexample to the theorem in respect to |G|+|H|. By Lemmas 2 and 4, it is clear that for any prime p dividing |H|, there exists a Sylow p-subgroup P of H such that every maximal subgroup of P is SE-supplemented in NH(P) and that P′ is S-quasinormal in H. Then H meets the hypothesis of Theorem 16, and thus H is supersolvable. Let q= max π(H) and Q∈Sylq(H); then Q⊴G. Let N be a minimal normal subgroup of G contained in Q, and consider the quotient group G/N. First, (G/N)/(H/N)=G/H∈ℱ. With a similar argument as in the proof of Theorem 16, we can obtain that G/N with subgroup H/N meets the hypothesis; hence G/N∈ℱ by the choice of G. Consequently, we may assume that H=Q=N is a minimal normal subgroup of G. Since ℱ constitutes a saturated formation, N≰Φ(G). Then there exists a maximal subgroup M of G such that G=NM and N∩M=1. Let Mq be a Sylow q-subgroup of M. Then Gq=QMq is a Sylow q-subgroup of G. Let Q1=Q∩Q0, where Q0 is a maximal subgroup of Gq containing Mq. Then Q1 is a maximal subgroup of Q and Q1⊴Gq. From this hypothesis, Q1 is SE-supplemented in NG(Q)=G. Let T be any supplement of Q1 in G; then Q1T=G and Q=Q∩Q1T=Q1(Q∩T). This means that Q∩T≠1. We note that Q∩T is normal in G and Q is a minimal normal subgroup of G, Q∩T=Q. thus T=G is the unique supplement of Q1 in G. As a result, we know that Q1=Q1∩T≤(Q1)sG by Lemma 6. Then (Q1)sG=Q1 is S-quasinormal in G. It follows that NG(Q1)≥Oq(G) by Lemma 5. As Q1⊴Gq, it is easy to see that NG(Q1)≥GqOq(G)=G. By the minimal normality of Q in G, it is clear that Q1=1 and thus Q is a cyclic group of order q. It follows that G∈ℱ by [6, Lemma 2.16], a contradiction. The proof is completed.From Theorem17 the following corollary is immediate.Corollary 18.
Letℱ be a saturated formation containing the class of all supersolvable groups 𝒰, and let G be a group with a normal subgroup H satisfying G/H∈ℱ. If for any prime p dividing |H|, there exists a Sylow p-subgroup P of H such that every maximal subgroup of P is S-quasinormally embedded or weakly SE-supplemented in NG(P) and such that also P′ is S-quasinormal in G, then G∈ℱ.



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