Finite p-nilpotent groups with some subgroups c-supplemented

J. Aust. Math. Soc. 78 (2005),
FINITE p-NILPOTENT GROUPS WITH SOME SUBGROUPS
c-SUPPLEMENTED
XIUYUN GUO and K. P. SHUM
(Received 10 June 2001; revised 18 February 2004) A subgroup H of a finite group G is said to be c-supplemented in G if there exists a subgroup K of Gsuch that G = H K and H ∩ K is contained in coreG .H /. In this paper some results for finite p-nilpotentgroups are given based on some subgroups of P c-supplemented in G, where p is a prime factor of theorder of G and P is a Sylow p-subgroup of G. We also give some applications of these results.
2000 Mathematics subject classification: primary 20D10, 20D20.
1. Introduction
Let G be a finite group. The relationship between the properties of the Sylow sub-groups of G and the structure of G has been investigated by a number of authors(see, for example, In particular, Buckley in 1970 provedthat a group of odd order is supersolvable if all its minimal subgroups are normal.
Srinivassan proved that a finite group is supersolvable if every maximal subgroupof every Sylow subgroup is normal. These two important results on supersolvablegroups have been generalized by many authors. One direction of generalization is toreplace the normality condition of maximal subgroups or minimal subgroups of Sylowsubgroups by a weaker condition; and the other direction of generalization is to min-imize the number of maximal subgroups or minimal subgroups of Sylow subgroups.
The research of the first author was partially supported by a grant of the Shanghai Natural ScienceFoundation (Grant No. 03ZR) and a grant of the National Natural Science Foundation of China (GrantNo. 10171086). The research of the second author is partially supported by a UGC(HK) grant #2160126(2003/2005).
c 2005 Australian Mathematical Society 1446-8107/05 $A2:00 + 0:00 It has been observed that the property of ‘normality’ for some maximal subgroups orsome minimal subgroups of Sylow subgroups gave a lot of useful information on thestructure of groups. In this paper, we shall continue to study the structure of finitegroups on the assumption that some subgroups are c-supplemented and obtain someinteresting results for finite p-nilpotent groups. As an application of our results, wegive conditions for a finite group to be in a saturated formation containing the class offinite supersolvable groups.
Throughout this paper, all groups are finite. Our terminology and notation are 2. Preliminaries
A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = H K and H ∩ K ≤ core .
several lemmas for later use in this paper.
LEMMA 2.1 Lemma 2.1]). Let H be a subgroup of a group G. Then the (1) Let K be a subgroup of G such that H is contained in K . If H is c-supplemented in G then H is c-supplemented in K . (2) Let N be a normal subgroup of G such that N is contained in H . Then H is c-supplemented in G if and only if H=N is c-supplemented in G=N . (3) Let ³ be a set of primes. Let N be a normal ³ -subgroup of G and H a ³- subgroup of G. If H is c-supplemented in G then H N =N is c-supplemented in G=N .
Furthermore, if N normalizes H , then the converse statement also holds. (4) Let L be a subgroup of G and H ≤ 8.L/. If H is c-supplemented in G then H is normal in G and H ≤ 8.G/. LEMMA 2.2 Lemma 2.6]). Let N be a solvable normal subgroup of a group G .N = 1/. If N ∩ 8.G/ = 1, then the Fitting subgroup F.N / of N is the directproduct of all minimal normal subgroups of G which are contained in N . Recall that a formation of groups is a class of groups homomorphic images and is such that G=M ∩ N ∈ subgroups of a group G with G=M ∈ Now we let 5 be the set of all prime numbers. Then, a function f defined on 5 is called a formation function if f . p/, possibly empty, is a formation for all p ∈ 5. Achief factor H=K of a group G is called f -central in G if G=C .
G H =K / ∈ f . p/ for all prime numbers p dividing |H=K |. A formation Finite p-nilpotent groups with some subgroups c-supplemented if there exists a formation function f such that which every chief factor of G is f –central in G. If a formation function f , then we write = L F. f / and we call f a local definition Among all the possible local definitions for a local formation there exists exactly one of them, denoted it by F, such that the formation function F isboth integrated (that is, F. p/ ⊆ for all p ∈ 5) and full (that is, Æp F. p/ = F. p/ for all p ∈ 5), where Æp is the class of p-groups.
LEMMA 2.3 Proposition IV. 3.11]). Let i is both an integrated and full formation function of the following statements are equivalent: 1 p/ ⊆ F2 p/ for all p ∈ 5. be a saturated formation. Assume that G is a group such that G does not belong to and there exists a maximal subgroup M and G = M F.G/, where F.G/ is the Fitting subgroup. Then G =.G / is a chief factor of G, G exponent p if p > 2 and exponent at most 4 if p = 2. Moreover, G elementary abelian group or .G / = Z .G / = 8.G / is an elementary abeliangroup. 3. Main results
We now establish our main theorems for p-nilpotent groups.
THEOREM 3.1. Let p be the smallest prime dividing the order of a group G and P a Sylow p-subgroup of G. If every minimal subgroup of P is c-supplemented in Gand when p = 2, either every cyclic subgroup of order 4 of P is also c-supplementedin G or P is quaternion-free, then G is p-nilpotent. PROOF. Suppose that the theorem is false and let G be a counterexample of the smallest order. Then G is not p-nilpotent. Since all Sylow p-subgroups of G areconjugate in G, we see that the hypotheses of our theorem is subgroup-closure byLemma Therefore G is a minimal non- p-nilpotent group (that is, every propersubgroup of a group is p-nilpotent but itself is not p-nilpotent). By a result of Itˆo Theorem 10.3.3], we know that G must be a minimal non-nilpotent group. Also by a result of Schmidt Theorem 9.1.9 and Exercises 9.1.11], we see that G is of order pÞqþ, where q is a prime distinct from p, P is normal in G and any Sylow q-subgroup Q of G is cyclic. Furthermore, P is of exponent p if p is odd and of exponent at most 4 if p = 2. Let A be a minimal subgroup of P. Then by our hypotheses, there existsa subgroup K of G such that G = AK and A ∩ K ≤ core .
If A is not normal in G then A ∩ K = 1 and therefore K is a maximal subgroup of G with index p. Since p is the smallest prime dividing the order of G, we seethat K is normal in G. Also since K is a proper subgroup of G, K is nilpotent. Itfollows that the Sylow q-subgroup of K is normal in G and therefore G is nilpotent,a contradiction. Hence, we may assume that every minimal subgroup of P must benormal in G and therefore every minimal subgroup of P must be in the center of G.
If p is odd, then G is p-nilpotent by Itˆo’s lemma, a contradiction. So there remainsthe case when p = 2.
Now let p = 2. By the above proof, we can see that every minimal subgroup of P lies in the center of G. If P is quaternion-free, then by applying Theorem 2.8], 1 P / ≤ P ∩ G Æ ∩ Z .G / = 1, where G Æ is the nilpotent residual of G , a contradiction. Now let every cyclic subgroup of order 4 of P be also c-supplementedin G and let B = b be a cyclic subgroup of P with order 4. Then, by our hypotheses,there exists a subgroup K of G such that G = B K and B ∩ K ≤ core .
lies in the center of G, we may replace K by K b2 if necessary and we may assumethat [G : K ] ≤ 2. If [G : K ] = 2, then K is normal in G and K is nilpotent. Sincethe normal p-complement of K is the normal p-complement of G, G is nilpotent, acontradiction. Hence, K = G and B must be normal in G. If B = P, then, sinceG is a minimal non-nilpotent group and the exponent of P is at most 4, we have G Q/ and therefore G = P × Q, a contradiction. If P = B, then it is clear that G is p-nilpotent, another contradiction. Thus, by all the above contradictions, weconclude that the theorem is true.
COROLLARY 3.2. Let N be a normal subgroup of a group G and p the smallest prime dividing the order of G. Also let be a saturated formation containing the class Æp of the all p-nilpotent groups and G=N ∈ P is c-supplemented in G, and when p = 2, either every cyclic subgroup of order 4 of P is also c-supplemented in G or P is quaternion-free, then G ∈ PROOF. It is easy to see from Lemma that every minimal subgroup of P is c-supplemented in N , and when p = 2 either every cyclic subgroup of order 4 of P is also c-supplemented in N or P is quaternion-free.
p-nilpotent. Let H be the normal p-complement of N . Then it is clear that H is normal in G and .G=H /=.N =H / Finite p-nilpotent groups with some subgroups c-supplemented satisfies the hypotheses of the corollary for normal subgroup N =H . Now if H = 1,by induction, we see that G=H ∈ . Let Fi (i = 1; 2) be the full and integrated = L F.F /, respectively. Then, it F1 q/ for every chief factor K1 K2 of G with K1 and every prime q dividing the order of |K = F2 q/ for every chief factor K1 K2 of G with K1 prime q dividing the order of |K = that H = 1 and N = P is a p-group. In this case, for any prime q dividing the orderof G with q = p and Q ∈ Syl .G/, it is clear that P Q is a subgroup of G and hence P Q is p-nilpotent by Theorem and therefore we have P Q = P × Q. It follows K2 is a p-group for every chief factor K1 K2 of G with K1 Now by using Lemma again, we see that G ∈ REMARK 3.3. The hypotheses that p is the smallest prime dividing the order of a group G in Theorem and Corollary cannot be removed. For example, G = S3,the symmetric group of order three, is an example for p = 3.
THEOREM 3.4. Let p be the smallest prime dividing the order of a group G and P a Sylow p-subgroup of G. If every maximal subgroup of P is c-supplemented in G,then G is p-nilpotent. PROOF. It is easy to see that every maximal subgroup of every Sylow p-subgroup of G is c-supplemented in G. Thus, in the following proof, we may make a choiceamong Sylow p-subgroups of G. Now, assume that the theorem is false and let Gbe a counterexample of minimal order. Then we prove the theorem by making thefollowing claims: G/ = 1, then we may choose a minimal normal subgroup N of G such G/. It is clear that P N =N is a Sylow p-subgroup of G=N . For every maximal subgroup P1 N =N of P N =N , we may assume that P1 is a maximal subgroupof P. Thus, by Lemma (3), every maximal subgroup of P N =N is c-supplementedin G=N . Hence, by the minimality of G, we know that G=N is p-nilpotent and so Gis p-nilpotent, a contradiction.
If G is odd, then G is solvable by the well-known odd order theorem of Feit and p G / = 1. Now let G be a group of even order and p G / = O2 G / = 1. Let P1 be a maximal subgroup of P . By hypotheses there exists a subgroup K of G such that G = P K = 1. Since [P : P1 it follows that the Sylow 2-subgroups of K are cyclic of order 2 and therefore K is2-nilpotent. Let K2 be the Hall 2 -subgroup of K . Then G = P K2 and K2 is a Hall 2 -subgroup of G. Assume that G is a non-abelian simple group. Then, by Corollary 5.6], G is isomorphic to PSL.2; r/ with r a Mersenne prime. In this case,by Corollary 5.8], every subgroup of G of 2-power index is the normalizer of aSylow r -subgroup of G. In particular, K and K2 have the same order, a contradiction.
Hence G is not a non-abelian simple group.
Let N be a minimal normal subgroup of G with N = G. Then N is neither a 2-group nor a 2 -group. Since G satisfies E2 (existence of Hall 2’-subgroups), weassume that N2 is a Hall 2 -subgroup of N and N2 a Sylow 2-subgroup of N . If P = N2, then N clearly satisfies the hypotheses of our theorem by Lemma (1).
Thus, by the minimality of G, we know that N is 2-nilpotent and hence O .
which contradicts to (1). On the other hand, if N2 is not a Hall 2 -subgroup of G,then P N is a proper subgroup of G and P N also satisfies the hypotheses of ourtheorem. Now, by the minimality of G again, P N is 2-nilpotent and therefore N itselfis 2-nilpotent. It follows that O .
G/ = 1, a contradiction again. Hence we conclude P and N2 is a Hall 2 -subgroup of G. Since G satisfies E2 , we can see that both G and N satisfy C2 (all Hall 2’-subgroups are conjugate) by Gross’ theorem Main Theorem]. Now by using the Frattini argument, we have Now let P∗ ∈ Syl .N .N // with P∗ ≤ P. Then, by our choice of G, we know G. Thus P ∗ < P and therefore there exists a maximal subgroup P1 of P such that P∗ ≤ P1. By our hypotheses again, there exists a subgroup K of G such that G = P K = 1. It is now clear that the order of Sylow 2-subgroups of K is 2 and therefore K is 2-nilpotent. Let H be a normal2-complement of K . Then, H is a Hall 2 -subgroup of G. Thus there exists anelement g of G such that H g = N2 . Since G = P1K and H is a normal subgroup of K , we may choose g ∈ P1. We also see that K g normalizes H g = N2 and thereforeK g ≤ N .
. Thus, it follows that G = Gg = .P1 K /g = P1NG N2 . This leads to P = P ∩ G = P .
p G // = 1, then we may consider the quotient group G=8.O p G //. Obvi- ously, by Lemma (2), every maximal subgroup of P=8.O .
p G //. Thus, by the minimality of G , we see that G=8.O p G // has a nor- mal p-complement T =8.O .
p G //. By the Schur-Zassenhaus Theorem, there exists a Hall p -subgroup H of T such that T = H 8.O .
p G //. By using the Frattini argument again, we see that G = 8.O .
p G // NG H / = NG H / since 8.O p G // ≤ 8.G /, a p G // = 1 and O p G / is an elementary abelian group.
p G / is a minimal normal subgroup of G .
Let N be a minimal normal subgroup of G such that N ≤ O .
Finite p-nilpotent groups with some subgroups c-supplemented is easy to see that G=N satisfies the hypotheses of our theorem. The minimalityof G implies that G=N is p-nilpotent. Similarly, if N1 is another minimal normalsubgroup of G with N ≤ Op G/, then we see that G=N1 is also p-nilpotent. Now G=N ∩ N1 is p-nilpotent, a contradiction. Hence N must be the unique minimal normal subgroup of G which is contained in O .
arguments similar to the proof in (3), we have G = N N .
p -subgroup of G. Since N .
G H / < G , it follows that N ≤ NG H / and then, since p G / ∩ NG H / is normal in G , O p G / ∩ NG H / = 1. Finally, by Dedekind’s law, p G / = N .O p G / ∩ NG H // = N . This proves (4).
(5) The final contradiction.
From the above proof, we see that G=O .
p G / is p-nilpotent. By using the arguments similar to the proof in (3), we have G = N N .
G H /, where H is a Hall p -subgroup of G. Let P∗ be a Sylow p-subgroup of N .
G H / < G and therefore P ∗ < P . Let P1 be a maximal subgroup of P with Since Op G/ is a minimal normal subgroup of G and Op G/ ≤ P1, we 1. By our hypotheses, there exists a subgroup K1 of G such that 1. It is clear that the order of Sylow p-subgroups of K1 is p and therefore K1 has a normal p-complement H1. Then there exists an elementg ∈ O .
p G / P ∗ such that . H1 H . Noticing that P1 is normal in Op G/P∗, we has normal p-complement and H = .H /g ≤ .
and consequently G = P .
P1 NG H /. Hence Op G/P∗ = .Op G/P∗/ ∩ p G / P ∗/ ∩ . P1 NG H // = P1 Op G/P∗/ ∩ NG H // = P1 P∗ = P1, which contradicts to the fact that O .
p G / P ∗ is a Sylow p-subgroup of G . Thus our theorem COROLLARY 3.5. If every maximal subgroup of every Sylow subgroup of a group G is c-supplemented in G, then G is a Sylow tower group of supersolvable type. PROOF. Let p be the smallest prime dividing the order of G and P a Sylow p- subgroup of G. By Corollary G is p-nilpotent. Let N be a normal p-complementof G. Clearly N satisfies the hypotheses of G and therefore by induction N is a Sylowtower group of supersolvable type. This proves that G is a Sylow tower group ofsupersolvable type.
By using the arguments similar to the proof in Corollary we can prove the COROLLARY 3.6. Let N be a normal subgroup of a group G and p the smallest prime dividing the order of G. Also let be a saturated formation containing the class Æp of the all p-nilpotent groups and G=N ∈ P is c-supplemented in G, then G ∈ , where P is a Sylow p-subgroup of N . 4. Applications
As an application of Theorem and Theorem we establish the following the- orems for a group to be in the saturated formation containing the class of supersolvablegroups.
be a saturated formation containing the class of supersolv- . Let N be a normal subgroup of a group G such that G=N is in for every prime p dividing the order of N and for every Sylow p-subgroup P of N ,every minimal subgroup of P is c-supplemented in G and when p = 2, either everycyclic subgroup of order 4 of P is also c-supplemented in G or P is quaternion-free,then G is in PROOF. Assume that the theorem is false and let G be a counterexample of minimal order. By Lemma and Theorem we know that N is a Sylow tower group ofsupersolvable type. Let q be the largest prime dividing the order of N and Q aSylow q-subgroup of N . Then Q is normal in G and every minimal subgroup of Qis c-supplemented in G. It is clear that .G=Q/=.N =Q/ satisfies the hypotheses of our theorem by Lemma The minimality of G impliesthat G=Q is in ≤ Q and G is a q-group, where G is the -residual of G. By Theorem 3.5], there exists a maximal subgroup M of G such that G = M F .G/, where F .G/ = Soc.G mod 8.G// and G=core .
and therefore G = M F.G/ since G F.G/ is the Fitting subgroup of G. It is now clear that M satisfies the hypotheses of our theorem for its normal subgroup M ∩ Q. Hence, by the minimality of G, it leadsto M must be in has exponent q when q = 2 and exponent at most 4 when is an elementary abelian group, then G of G. For any minimal subgroup A of G , we know that A is c-supplemented in Gby our hypotheses. Hence there exists a subgroup K of G such that G = AK and G A/. If A is not normal in G , then A ∩ K = 1. It is clear that K ∩ G is normal in G. The minimality of G in G, a contradiction. Hence A is normal in G and G = A is cyclic of order q.
is not an elementary abelian group, then .G / = Z .G / = 8.G / is an elementary abelian group by Lemma Noticing that 8.G / ≤ 8.G/, we knowthat every minimal subgroup of .G / is not complemented in G. It now followsfrom our hypotheses that every minimal subgroup of .G / must be normal in G.
Finite p-nilpotent groups with some subgroups c-supplemented For any minimal subgroup A of G =.G / , there exists a subgroup A of G that A = A.G / =.G / . Assume that A is of order q. If A is not normal in G,then, by our hypotheses, there exists a subgroup K of G such that G = AK and A ∩ K = 1. Noticing that .G / = 8.G / ≤ 8.G/, we see that K =.G / is a complement of A. The minimality of G =.G / implies that A = G =.G / isnormal in G=.G / , and therefore G =.G / is a cyclic group of order q. Hence wemay assume that q = 2 and every generated element of G / = .G / = 8.G / and therefore every minimal subgroup / is normal in G. Hence 1 G / ≤ Z.G/. If Q is quaternion-free, then, by Lemma 2.15], every 2 -element of G acts trivially on G . Since G =.G /is a chief factor of G, we see that G =.G / is a cyclic group of order 2. Assumethat every cyclic group of order 4 of Q is c-supplemented in G. Let B = b bea cyclic group of order 4 of G . Then b2 is normal in G. If B is not normal inG, then there exists a subgroup of K of G such that G = B K and B ∩ K = b2 .
It is clear that .G / = 8.G / ≤ K and G =.G / ∩ K =.G / is normal inG=.G / . The minimality of G =.G / implies that G =.G / ∩ K =.G / = 1and therefore G =.G / is a cyclic group of order 2. We have now shown that for allcases, G =.G / is always a cyclic group of prime order. Noticing that G =.G /is G-isomorphic to Soc.G=core .
G M //, it follows that G=coreG M / is supersolvable, a contradiction. Thus, our proof is completed.
be a saturated formation containing the class of supersolv- . Let N be a normal subgroup of a group G such that G=N is in for every prime p dividing the order of N and for every Sylow p-subgroup P of N ,every maximal subgroup of P is c-supplemented in G, then G is in i i = 1; 2/ be the full and integrated formation functions such that = L F.F /. Assume that the theorem is false and we may let G be a minimal counterexample. Then, by applying Lemma and Corollary we know that N has a Sylow tower of supersolvable type. Let p be the largest primedividing the order of N and P ∈ Syl .N /. Then P must be a normal subgroup of G. Clearly, .G=P/=.N =P/ . It is easy to see that G=P satisfies our hypotheses of the theorem for the normal subgroup N =P. By the minimality of G, wesee that G=P ∈ , and of course, every maximal subgroup of P is c-supplemented Let L be a minimal normal subgroup of G with L ≤ P. Then, it is easy to see that the quotient group G=L satisfies the hypotheses of our theorem for the normalsubgroup of P=L. By our choice of G, we have G=L ∈ formation, L is the unique minimal normal subgroup of G which is contained in Pand also L is complemented in G. In particular, we have P ∩ 8.G/ = 1 and therefore L = F.P/ = P is an abelian minimal normal subgroup of G by Lemma Let P1 be a maximal subgroup of P. By our hypotheses, there exists a subgroup K of G such that G = P K = 1 since L is the unique minimal normal subgroup of G contained in P with L ≤ P 1. Thus P = P1 P ∩ K /. It is clear that P ∩ K is normal in K and is normalized by P1 since P is abelian. Therefore P ∩ K is a normal subgroup of G. Since P ∩ K = 1 and P is a minimal normal subgroup of G,it follows that P ∩ K = P and P is a cyclic group of order p. Since Aut .P) is a cyclicgroup of order p − 1 and G=C .
G P / ≤ Aut. P /, we have G=CG P / ∈ F1 p/ ⊆ F2 p/, , a contradiction. Thus, our proof is completed.
be the class of groups G whose derived group G is nilpotent.
is a saturated formation containing the class by applying our Theorems and we also obtain some sufficient conditions fora group to be a Acknowledgement
The authors would like to thank the referee for his comments contributed to the References
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