 Applied Mathematics, 2011, 2, 562-564 doi:10.4236/am.2011.25074 Published Online May 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Two Theorems about Nilpotent Subgroup Lijiang Zeng Department of Mat hematics, Zunyi Nor m al C oll e ge, Zu nyi , Chin a E-mail: zlj4383@sina.com Received February 3, 2011; revised March 21, 2011; acce p ted Mar ch 23, 2011 Abstract In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theo-rems about nilpotent subgroup. Keywords: Hall π-Subgroup, Sylow p-Subgroup, Normalizer, Nilpotent Group In this paper, we introduced some concepts and notations such as Hall π-subgroup and so on. Using concepts, terms and notations in group theory, we have proved some properties about finite group, nilpotent group and Sylow p-subgroup, and proved two interesting theorems about nilpotent subgroup in these properties. Let π be a set of some primes and the supplementary set of π in the set of all primes be notated π', When π contains only one prime p we notate π and π' as p and p', When all prime factor of integer n be in π we called n as a π-number, If the order H of G’s subgroup be a π-number we called H as a π-subgroup. Definition 1. If H be a π-subgroup of G and :GH be a π'- number, we called H as a Hall π-subgroup of G. Lemma 1. A nontrivial finite p-group has a nontrivial center. Proof. Let 1 be the class equation  of the group; ni divides pm and hence is a power of p. If the center were trivial, only ni would equal 1 and , which is impossible since . mkpn np1modmp1mpDefinition 2. A group G is called nilpotent  if it has a central series , that is, a normal series 011nGG GG such that iGG is contained in the center of iGG for all i. The length of a shortest central series of G is the nilpotent class of G. A nilpotent group of class 0 has order l of course, while nilpotent groups of class at most 1 are abelian. Whereas nilpotent groups are obviously soluble, an ex-ample of a non nilpotent soluble group is 3 (its centre is trivial). The great source of finite nilpotent groups is the class  of groups whose orders  are prime pow-ers. SLemma 2. A finite p-group is nilpotent. Proof. Let G be a finite p-group of order > 1. Then Lemma 1 shows that 1G. Hence GG is nilpo-tent by induction on G. By forming the preimages of the terms of a central series of GG under the natural homomorphism  GGG and adjoining  1, we arrive at a central series of G. Lemma 3.The class of nilpotent groups is closed un-der the formation of subgroups, images, and finite direct products. The proof can be found in Reference . Lemma 4. Let P be a Sylow p-subgroup  of a finite group G. i) If GNP HG, then GHNH. ii) If , then NGPN is a Sylow p-subgroup of N and PNN is a Sylow p-subgroup of GN. Proof. i) Let GxNH. Since GPHNH, we have xPH. Obviously P and Px are Sylow p-sub-group of H, so xhPP for some . Hence hHG1xhNPH and xH. It follows that GHNH. ii) In the first place ::PNPNP N , which is prime to p. Since PN is a p-subgroup, it must be a Sylow p-subgroup of N. For PNN the argument is similar. Lemma 5. Let G be a finite group. Then the following properties are equivalent: Foundation Item: Project supported by Natural Science Foundation(13116339) of China; Natural Science Foundation (2075) of Sci-ence and Technology Department of Guizhou; Natural Science Founda-tion (069) of Education Department of Guizhou; Science Re-search item(2010028) of Zunyi Normal College. i) G is nilpotent; ii) every subgroup of G is subnormal ; iii) G satisfies the normalizer  condition; L. J. ZENG563 iv) every maximal subgroup  of G is normal; v) G is the direct product of its Sylow subgroups. Proof: i)→ii) Let G be nilpotent with class c. If HG, then 1iiHGH G since 1ii iGG GG. Hence 01 cHHGHG HGG  and H is subnormal in G in c steps. ii)→iii) Let HG. Then H is subnormal in G and there is a series 01nHHH HGi. If i is the least positive integer such that HH. Then 1iiHHH and iGHNH. iii)→iv) If M is a maximal subgroup of G, then GMNM, so by maximality GNM G and MG. iv)→v) Let P be a Sylow subgroup of G. If P is not normal in G, then is a proper subgroup of G and hence is contained in a maximal subgroup of G, say M. Then GNPMG; however this contradicts Lemma 4. Therefore each Sylow subgroup of G is normal and there is exactly one Sylow p-subgroup for each prime p since all such are conjugate. The product of all the Sylow sub-groups is clearly direct and it must equal G. v)→i) by Lemma 2 and Lemma 3. Theorem 1. Assume that every maximal subgroup of a finite group G itself is not nilpotent. Then: i) G is soluble; ii) mnGpq where p and q are unequal primes; iii) there is a unique Sylow p-subgroup P and a Sylow q-subgroup Q is cyclic. Hence and . GQPPGProof. i) Let G be a counterexample of least order. If N is a proper nontrivial normal subgroup, both N and GNare soluble, whence G is soluble. It follow that G is a simple group. Suppose that every pair of distinct maximal subgroups of G intersects in 1. Let M be any maximal subgroup: then certainly GMNM If Gn and Mm, then M has nm conjugates  every pair of which intersect trivially. Hence the conjugates of M account for exactly 1mn nnmm nontrivial elements. Since m ≥ 2, we have 122nnnnm in addition it is clear that 21. nnnnmSince each nonidentity element of G belongs to ex-actly one maximal subgroup, n – 1 is the sum of integers lying strictly between 12n and n – 1. This is plainly impossible. It follows that there exist distinct maximal subgroups M1 and M2 whose intersection I is nontrivial. Let M1 and M2 be chosen so that I has maximum order . Write GNNI. Since M is nilpotent, 1MINI by Lemma 5, so that 1INM. Now I cannot be normal in G; thus N is proper and is contained in a maximal subgroup M. Then 11INM MM, which con-tradicts the maximality of I. ii) Let 11kepeeGpk, where and the i are distinct primes. Assume that . If M is a maxi-mal normal subgroup, its index is prime since G is solu-ble; let us say 0i3pk1:GMp. Let i be a Sylow pi-subgroup of G. If , then i and, since M is nilpotent, it follows that i; also the since . Hence P1Pi is nilpotent and thus PM1iPGP3k1,iPP1 (by Lemma 5). It follows that 1NPG and 1. This means that all Sylow subgroup of G are normal, so G is nilpotent. By this contradiction k = 2 and GPG122ep1eGp. We shall write 2pp and q. 1iii) Let there be a maximal normal subgroup M with index  q. Then the Sylow p-subgroup P of M is nor-mal in G and is evidently also a Sylow p-subgroup of G. Let Q be a Sylow q-subgroup of G. Then G = QP. Sup-pose that Q is not cyclic. If pgQ, then ,gPG since otherwise QGP, which is cyclic . Hence ,gP is nilpotent and ,1gP. But this means that ,PQ 1 and GPQ, a nilpotent group. Hence Q is cyclic. In an insoluble group  Hall π-subgroups, even if they exist, may not be conjugate: for example, the simple group PSL (2, 11) of order 660 has subgroups isomor-phic with D12 and A4: these are nonisomorphic  Hall 2,3 -subgroups and they are certainly not conjugate. However the situation is quite different when a nilpotent Hall π-subgroup is present. Theorem 2. Let the finite group G possess a nilpotent Hall π-subgroup H. Then every π-subgroup of G is con-tained in a conjugate of H. In particular all Hall π-subgroups of G are conjugate. Proof. Let K be a π-subgroup of G. We shall argue by induction on K, which can be assumed greater than l. By the induction hypothesis a maximal subgroup of K is contained in a conjugate of H and is therefore nilpotent. If K itself is not nilpotent, Theorem 1 may be applied to produce a prime q in π dividing K and a Sylow q-subgroup Q which has a normal complement L in K. Of course, if K is nilpotent, this is still true by Lemma 5. Now write 12HHH where H1 is the unique Sy-low q-subgroup of H. Since , the induction hy-LKCopyright © 2011 SciRes. AM L. J. ZENG Copyright © 2011 SciRes. AM 564  C. W. Curtis and I. Reiner, “Methods of Representation Theory,” Wiley, New York, 1981, pp. 177-184. pothesis shows that 12gggLHH H  J. E. Roseblade, “A Note on Subnormal Coalition Cla- sses,” Mathematische Zeitschrift, Vol. 90, No. 5, 1965, pp. 373-375. doi:10.1007/BF01112356 for some gG. Thus 2gLH because L is a q'-group. Consequently GNNL contains 1,gHK. Ob-serve  C. F. Miller III, “On Group Theoretic Decision Problems and Their Classification,” Annals of Mathematics Studies, No. 68, Princeton University Press, Princeton, 1971, pp. 112-134. that 1:H is not divisible by q; hence 1GgH is a Sylow q-subgroup of N and by Sylow’s Theorem 1xgQH for some xN. But xLL and, using 2gLH, we obtain  M. F. 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