Math-Net.Ru All Russian mathematical portal Wenbin Guo, Alexander N. Skiba, On the intersection of maximal supersoluble subgroups of a finite group, Tr. Inst. Mat., 2013, Volume 21, Number 1, 48 51 Use of the all-russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of use http://www.mathnet.ru/eng/agreement Download details: IP: 37.44.193.209 January 27, 2018, 12:35:37
Национальная академия наук Беларуси Труды Института математики. 2013. Том 21. 1. С. 48 51 УДК 512 542 ON THE INTERSECTION OF MAXIMAL SUPERSOLUBLE SUBGROUPS OF A FINITE GROUP Wenbin Guo Alexander N. Skiba Department of Mathematics, University of Science and Technology of China, Гомельский государственный университет им. Ф. Скорины e-mail: wbguo@ustc.edu.cn, alexander.skiba49@gmail.com Поступила 11.01.2013 Посвящается памяти Леонида Алексанровича Шеметкова The hyper-generalized-center genz (G) of a finite group G coincides with the largest term of the chain of subgroups 1 = Q 0 (G) Q 1 (G)... Q t (G)... where Q i (G)/Q i 1 (G) is the subgroup of G/Q i 1 (G) generated by the set of all cyclic S -quasinormal subgroups of G/Q i 1 (G). It is proved that for any finite group A there is a finite group G such that A G and genz (G) = Int (G). Throughout this note, all groups are finite and G always denotes a finite group. We use N and U to denote the classes of all nilpotent and of all supersoluble groups, respectively. A subgroup H of G is said to be S -permutable, S -quasinormal, or π -quasinormal [1] in G if HP = P H for all Sylow subgroups P of G. The hyper-generalized-center genz (G) of G coincides with the largest term of the chain of subgroups 1 = Q 0 (G) Q 1 (G)... Q t (G)... where Q i (G)/Q i 1 (G) is the subgroup of G/Q i 1 (G) generated by the set of all cyclic S -quasinormal subgroups of G/Q i 1 (G) (see [2, p. 22]). Let X be a class of groups. A subgroup U of G is called X -maximal in G provided that (a) U X and (b) if U V G and V X then U = V [3, p. 288]. Following [4, 5], we denote the intersection of all X -maximal subgroups of G by Int X (G). It is clear that the intersection of all maximal abelian subgroups of a group G coincides with the center Z(G) of G. In the paper [6], Baer proved that Int N (G) coincides with the hypercentre Z (G) of G. Further, Agrawal proved [7] that the subgroup genz (G) is contained in Int U (G) and posed the following question: Whether genz (G) = Int U (G) for any group G? (see [7, p. 19] or [2, p. 22]) In the papers [8, 9] it was proved, independently, that the answer to this question is negative. In this note we prove the following general result in this trend. Theorem. For any group A there is a group G such that A G and genz (G) = Int U (G). 48
We will use In the proof of this theorem the following known results. Lemma 1 (See Theorem C in [9]). Let H be a subgroup of G N a normal subgroup of G and I = Int (G). a) Int (H)N/N Int (HN/N). b) If H U then IH U. Lemma 2. Let H be a subgroup of G. a) If H is S -quasinormal in G then H is subnormal in G [1]. b) Suppose that H a p -subgroup of G for some prime p. Then H is S -quasinormal in G if and only if O p (G) N G (H) [10]. Lemma 3. Let H be a subnormal subgroup of G. a) If H is nilpotent, then H F (G) [11]. b) If H is a π -group, then A O π (G) [11]. Proof of Theorem. Let A be any group. Then there is a prime p such that p > A and p 1 is not a prime power. Indeed, if p is a prime with p > A then, by the classical Chebyshev s theorem, there is a prime q such that p < q < 2p. It is also clear that at lest one of the numbers p and q is not of form r n + 1 where r is a prime. Let C p be a group of prime order p. Then π(aut(c p )) > 1. Let R and L be Hall subgroups of Aut(C p ) such that Aut(C p ) = R L and for any r π(r) and q π(l) we have r < q. Let B = (C p R) L = K L be the regular wreath product of C p R with L where K is the base group of G. Let P = C p (we use here the terminology in [3, Chapter A]). Then by Proposition 18.5 in [3, Chapter A], B is a primitive group and P is a unique minimal normal subgroup of B. Hence P = F (B) = C B (P ) by Theorem 15.2 in [3, Chapter A]. Moreover, by Lemma 18.2 in [3, Chapter A], B = P M where M U = R L = D L where D is the base group of U. It is clear that D is an abelian Hall subgroup of U and L is a cyclic subgroup of U such that r < q for any r π(d) and q π(l). Moreover, since Aut(C p ) = p 1 D and L are groups of exponent dividing p 1. First we show that every supersoluble subgroup W of U is nilpotent. Suppose that this is false and let H be a Schmidt subgroup of W. Then 1 < D H < H where D H is a normal Hall subgroup of H. By Theorem 26.3 in [12, Chapter VI], there are primes r and q such that H = H r H q where H r is a Sylow r -subgroup of H H q is a cyclic Sylow q -subgroup of H. Hence D H = H r. Since H W H is supersoluble and hence r > q. But Q H/D H HD/D isomorphic to some subgroup of L so r < q. This contradiction shows that W is nilpotent. Now we show that P Int U (B). Let V be any supersoluble subgroup of B and W a Hall p -subgroup of V. Then P V = P W. It is clear that M is a Hall p -subgroup of B. Hence for some x B we have W M x U x. Hence W is nilpotent. It is clear that the Sylow subgroups of W are abelian. Hence W is an abelian group of exponent dividing p 1. It follows from Lemma 4.1 in [12, Chapter I] that P V is supersoluble. Therefore P Int (B). Next we show that genz (B) = 1. Indeed, suppose that genz (B) = 1. Then B has a non-identity cyclic S -quasinormal subgroup, say V. The subgroup V is subnormal in B by Lemma 2 (a). Therefore V F (B) = P by Lemma 3 (a). Moreover, if Q is a Sylow q -subgroup of B where q = p then V is subnormal in V Q and so Q N B (V ). Hence V is normal in B and therefore V = P is cyclic. But then P = p = Cp a contradiction. Thus genz (B) = 1. 49
Finally, let G = A B and I = Int U (G). We will prove that I = genz (G). First we show that P I. Let V be a supersoluble subgroup of G. By Lemma 1 (a), AP/A Int U (G/A). On the other hand, is supersoluble. Hence V A/A V/V A (P A/A)(V A/A) = P V A/A P V/P V A is supersoluble by Lemma 1 (b). Moreover, P V B/B P V/P V B = P V/P (V B) = P (V B)V/P (V B) = V/V B is supersoluble. Therefore is supersoluble. This implies that P I. Now let P V P V/1 = P V/(P V A) (P V B) 1 = Q 0 Q 1... Q t = genz (G) where Q i /Q i 1 is the subgroup of G/Q i 1 generated by the set of all cyclic S -quasinormal subgroups of G/Q i 1. Suppose that I = genz (G). Then P genz (G) and so t > 0. Since P is a minimal normal subgroup of B P is also a minimal normal subgroup of G. Hence there is a number i such that P Q i 1 and P Q i. Let H/Q i 1 be any cyclic S -quasinormal subgroup of G/Q i 1. Then H/Q i 1 is subnormal in G/Q i 1 by Lemma 2 (a). Hence H/Q i 1 O π (G/Q i 1 ) where π = π(h/q i 1 ) by Lemma 3 (a). It follows that some subgroup V/Q i 1 of G/Q i 1 of order p is S -quasinormal in G/Q i 1. Since P Q i 1 /Q i 1 is a Sylow p -subgroup of G/Q i 1 and P is abelian, V/Q i 1 is normal in G/Q i 1 by Lemma 2 (b). By the Schur Zassenhaus theorem, V = Q i 1 C p for some subgroup C p of V of order p. Since I = genz (G) V is supersoluble. On the other hand, by the choice of p p is the largest prime dividing V. Hence C p is normal in V and so C p is characteristic in V. It follows that C p is normal in G. Consequently C p genz (B) = 1. This contradiction completes the proof of the result. References 1. Kegel O.H. Sylow-Gruppen and Subnormalteiler endlicher Gruppen // Math. Z. 1962. V. 78. P. 205 221. 2. Weinstein M. Between Nilpotent and Solvable. Polygonal Publishing House. 1982. 3. Doerk K. Hawkes T. Finite Soluble Groups. Berlin; New York: Walter de Gruyter. 1992. 4. Skiba A.N. On the F -hypercentre and the intersection of all F -maximal subgroups of a finite group // Journal of Pure and Applied Algebra. 2012. V. 216. P. 789 799. 5. Guo W. Skiba A.N. On the intersection of the F -maximal subgroups and the generalized F -hypercentre od a finite group // J. Algebra. 2012. V. 366. P. 112 125. 50
6. Baer R. Group elements of prime power index // Trans. Amer. Math. Soc. 1953. V. 75. P. 20 47. 7. Agrawal R.K. Generalized center and hypercenter of a finite group // Proc. Amer. Math. Soc. 1976. V. 54. P. 13 21. 8. Beidleman J.C. Heineken H. A note on intersections of maximal F -subgroups // J. Algebra. 2010. V. 333. P. 120 127. 9. Skiba A.N. On the intersection of all maximal F -subgroups of a finite group // J. Algebra. 2011. V. 343. P. 173 182. 10. Schmid P. Subgroups Permutable with All Sylow Subgroups // J. Algebra. 1998. V. 82. P. 285 293. 11. Wielandt H. Subnormal subgroups and permutation groups. Lectures given at the Ohio State University, Columbus, Ohio. 1971. 12. Shemetkov L.A. Formations of Finite Groups. Moscow: Nauka. 1978. Wenbin Guo, Alexander N. Skiba On the intersection of maximal supersoluble subgroups of a finite group Summary The hyper-generalized-center genz (G) of a finite group G coincides with the largest term of the chain of subgroups 1 = Q 0 (G) Q 1 (G)... Q t (G)... where Q i (G)/Q i 1 (G) is the subgroup of G/Q i 1 (G) generated by the set of all cyclic S -quasinormal subgroups of G/Q i 1 (G). It is proved that for any finite group A there is a finite group G such that A G and genz (G) = Int (G). 51