Influence of strongly closed 2-subgroups on the structure of finite groups more

INFLUENCE OF STRONGLY CLOSED 2-SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS HUNG P. TONG-VIET Abstract. Let H ≤ K be subgroups of a group G. We say that H is strongly closed in K with respect to G if whenever ag ∈ K, where a ∈ H, g ∈ G, then ag ∈ H. In this paper, we investigate the structure of a group G under the assumption that every subgroup of order 2m (and 4 if m = 1) of a 2Sylow subgroup S of G is strongly closed in S with respect to G. Some results related to 2-nilpotence and supersolvability of a group G are obtained. This is a complement to Guo and Wei (J. Group Theory 13 (2010), no. 2, 267–276). 1. Introduction All groups are finite. Let H ≤ K be subgroups of a group G. We say that H is strongly closed in K with respect to G if whenever a ∈ H, ag ∈ K, where g ∈ G then ag ∈ H. We also say that H is strongly closed in G if H is strongly closed in NG (H) with respect to G. The structure of groups which possess a strongly closed p-subgroup has been extensively studied. One of the most interesting results is due to Goldschmidt [4] which classified groups with an abelian strongly closed 2subgroup. This result is a generalization of the celebrated Glauberman Z ∗ -theorem. These results play an important role in the proof of the classification of the finite simple groups. Recently, Bianchi et al. in [3], called a subgroup H, an H-subgroup of G if H g ∩ NG (H) ≤ H for all g ∈ G. It is easy to see that these two definitions coincide. With this concept, they gave a new characterization of supersolvable groups in which normality is a transitive relation which are called supersolvable T -groups. In more detail, it is shown that every subgroup of G is strongly closed in G if and only if G is a supersolvable T -group (see [3, Theorem 10]). Some local versions of this result have been studied in [1] and [7]. For example, Asaad ([1, Theorem 1.1]) proved that G is p-nilpotent if and only if every maximal subgroup of a p-Sylow subgroup P of G is strongly closed in G and NG (P ) is p-nilpotent. Guo and Wei ([7, Theorem 3.1]) showed that whenever p is odd and P is a p-Sylow subgroup of G, G is p-nilpotent if and only if NG (P ) is p-nilpotent and either P is cyclic or every nontrivial proper subgroup of a given order of P is strongly closed in G. Also these results still hold without the p-nilpotence assumption on NG (P ) if p is the smallest prime divisor of the order of G. The purpose of this paper is to prove the following theorem, which is a complement to [7, Theorem 3.1]. Date: April 26, 2011. 2000 Mathematics Subject Classification. Primary 20D20. Key words and phrases. strongly closed, p-nilpotent, supersolvable. Support from the Leverhulme Trust is acknowledged. 1 2 HUNG P. TONG-VIET Theorem 1.1. Let P ∈ Syl2 (G) and D ≤ P with 1 < |D| < |P |. If P is either cyclic or every subgroup of P of order |D| (and 4 if |D| = 2) is strongly closed in G, then G is 2-nilpotent. The following example shows that the additional assumption when |D| = 2 in Theorem 1.1 is necessary. Example. Let G = SL2 (17). Then if P ∈ Syl2 (G) then P ∼ Q32 , a quaternion = group of order 32. Moreover P is maximal in G and hence NG (P ) = P is 2-nilpotent in G. Clearly, the center of G is a unique subgroup of order 2 and so it is strongly closed in G. However G is not 2-nilpotent. Theorem 1.1 above and [7, Theorem 3.4] now yield: Theorem 1.2. Let p be the smallest prime divisor of |G| and P ∈ Sylp (G). If P is cyclic or P has a subgroup D with 1 < |D| < |P | such that every subgroup of P of order |D| (and 4 if |D| = 2) is strongly closed in G, then G is p-nilpotent. We can now drop the odd order assumption on Theorems 3.5 and 3.6 in [7]. Theorem 1.3. If every non-cyclic Sylow subgroup P of G has a subgroup D with 1 < |D| < |P | such that every subgroup of P of order |D| (and 4 if |D| = 2) is strongly closed in G, then G is supersolvable. Theorem 1.4. Let E be a normal subgroup of G such that G/E is supersolvable. If every non-cyclic Sylow subgroup P of E has a subgroup D with 1 < |D| < |P | such that every subgroup of P of order |D| (and 4 if |D| = 2) is strongly closed in G, then G is supersolvable. 2. Preliminaries In this section, we collect some results needed in the proofs of the main theorems. Lemma 2.1. (Schur-Zassenhauss [6, Theorem 6.2.1]). If P is a normal 2-Sylow subgroup of G then G possesses a complement Hall-2 -subgroup. Lemma 2.2. ([6, Theorem 7.6.1]). If a 2-Sylow subgroup of G is cyclic then G is 2-nilpotent. Lemma 2.3. ([1, Corollary 1.2]). Let P be a 2-Sylow subgroup of G. Then G is 2-nilpotent if and only if every maximal subgroup of P is strongly closed in G. Lemma 2.4. Suppose that H is a strongly closed p-subgroup of G. (a) If H ≤ L ≤ G then H is strongly closed in L; ¯ ¯ ¯ (b) If G is a homomorphic image of G, then H is strongly closed in G and ¯ = NG (H); NG (H) ¯ (c) If H is subnormal in G then H G. Proof. (a) is [3, Lemma 7(2)] and (c) is [3, Theorem 6(2)]. Finally (b) is [5, (2.2)(a)]. Lemma 2.5. ([5, Corollary B3]). Suppose that H is a strongly closed 2-subgroup of G and NG (H)/CG (H) is a 2-group. Then H ∈ Syl2 ( H G ). Lemma 2.6. ([8, Satz 4.5.5]). If every element of order 2 and 4 of G are central then G is 2-nilpotent. STRONGLY CLOSED 2-SUBGROUPS 3 Lemma 2.7. ([2, Baumann]). If G is a non-abelian simple group in which a 2Sylow subgroup of G is maximal, then G is isomorphic to L2 (q), where q is a prime number of the form 2m ± 1 ≥ 17. A component of G is a subnormal quasisimple subgroup of G. Denote by E(G) the subgroup of G generated by all components of G. Then the generalized Fitting subgroup F ∗ (G) of G is a central product of E(G) and the Fitting subgroup F (G) of G. Lemma 2.8. ([9, Theorem 9.8]). CG (F ∗ (G)) ≤ F ∗ (G). Lemma 2.9. ([9, Problem 4D.4, p. 146]). Let A act via automorphisms on a 2-group P, where |A| is odd. If A centralizes every element of order 2 and 4 in P, then A acts trivially on P. The following result is a special case of [7, Lemma 2.10]. Lemma 2.10. Let P be an elementary abelian 2-subgroup of G and D a subgroup of P with 1 < |D| < |P |. If every subgroup of P of order |D| is normal in G, then every minimal subgroup of P is central in G. Proof. It follows from [7, Lemma 2.10] that every minimal subgroup of P is normal in G. As minimal subgroups of P are cyclic of order 2, they are all central. Lemma 2.11. Let A be an odd order group acting on a 2-group P. Let D ≤ P with 1 < |D| < |P |. If every subgroup of P of order |D| (and 4 if |D| = 2) is A-invariant, then A acts trivially on P. Proof. We can assume that |D| ≥ 4. Let D = {E ≤ P : |E| = |D|}. Suppose that D < P. If | D | > |D|, then by inductive hypothesis, A centralizes D , so that it centralizes every subgroup of P of order 2 and 4, hence the result follows from Lemma 2.9. If |D| = |D|, then P has a unique subgroup of order |D|. As 2 < |D| < |P |, P must be cyclic and thus A centralizes P by applying Lemma 2.2 to the semidirect product A P. Therefore, we can assume that D = P. Next, if A centralizes every element of D, then as |D| ≥ 4, A centralizes every element of order 2 and 4, and we are done by using Lemma 2.9. Hence there exists E ∈ D such that [E, A] = 1. It follows that Φ(P ) ≤ E, otherwise, E < EΦ(P ) < P, and by applying the inductive hypothesis for EΦ(P ), A would centralize E, which contradicts the choice of E, thus prove the claim. If Φ(P ) is trivial, then P is elementary abelian, and hence the result follows from Lemma 2.10. Thus Φ(P ) > 1. Assume that |E/Φ(P )| ≥ 2. By Lemma 2.10 again, A centralizes P/Φ(P ), and then [P, A] ≤ Φ(P ). By Coprime Action Theorem, A acts trivially on P and we are done. Thus we assume that E = Φ(P ). For any F ∈ D − {Φ(P )}, we have |F | = |Φ(P )| and Φ(P ) = F, it follows that F < F Φ(P ) < P and F Φ(P ) is A-invariant. By inductive hypothesis, A centralizes F, and hence P, as P is generated by D−{Φ(P )}. The proof is now complete. 3. Proofs of the main results Proposition 3.1. Let P ∈ Syl2 (G) and D ≤ P with 2 < |D| < |P |. Assume that either P is cyclic or every subgroup of P of order |D| is strongly closed in G, then G is 2-nilpotent. 4 HUNG P. TONG-VIET Proof. Suppose that the proposition is false. Let G be a minimal counter example. By Lemma 2.2, we can assume that P is non-cyclic. ¯ Claim 1. O2 (G) = 1. Assume that O2 (G) = 1. Passing to G = G/O2 (G), we ¯ satisfies the hypothesis of the proposition by Lemma 2.4(b), so that by see that G ¯ inductive hypothesis, G is 2-nilpotent and hence G is 2-nilpotent. Claim 2. If L G and L = G, then L ≤ O2 (G). Assume that L is a proper normal subgroup of G which is not a 2-group. As L G, P L is a subgroup of G. Assume that P L = G. By Lemma 2.4(a) and the inductive hypothesis, P L is 2-nilpotent. Let Q = O2 (P L). Then 1 = Q ≤ L G and since Q is characteristic in L, we have Q G and hence Q ≤ O2 (G) = 1 by Claim 2, which is a contradiction. Thus G = P L. Let U = P ∩ L. Then U ∈ Syl2 (L). Suppose that U is not maximal in P. Let P1 be a maximal subgroup of P that contains U. By comparing the order, we see that P1 L is a proper subgroup of P L = G. Then by Lemma 2.3, 2 < |D| < |P1 | and so P1 L is 2-nilpotent by induction. Arguing as above, we obtain 1 = O2 (P1 L) ≤ L G and hence O2 (P1 L) ≤ O2 (G) = 1. This contradiction shows that U is maximal in P. Now by Lemma 2.3 again, 2 < |D| < |U |. By induction again, L is 2-nilpotent which leads to a contradiction as above. This proves our claim. Claim 3. NG (P ) is 2-nilpotent. If NG (P ) < G, then it is 2-nilpotent by induction and we are done. Thus assume that NG (P ) = G. Then P G and hence every subgroup of P of order |D| is both subnormal and strongly closed in G so that they are normal in G by Lemma 2.4(c). By Schur-Zassenhaus Theorem, there exists a subgroup A of odd order such that G = P A. Since every subgroup of P of order |D| with 2 < |D| < |P | is A-invariant, by Lemma 2.11, A centralizes P and hence G is 2-nilpotent, which contradicts our assumption. Claim 4. F ∗ (G) = O2 (G). As O2 (G) = 1, we have F ∗ (G) = O2 (G)E(G). Assume that E(G) = 1. By Claim 2, we have E(G) = G and then by applying that claim again, we see that G must be a quasisimple group. Let H ≤ P be any subgroup of order |D|. Assume first that H ≤ Z(G). Then H is not normal in G so that H G = G and P ≤ NG (H) < G. By induction NG (H) is 2-nilpotent so that NG (H)/CG (H) is a 2-group. By Lemma 2.5, H ∈ Syl2 (G), which is a contradiction as |H| < |P |. Thus H ≤ Z(G) and since |D| > 2, every subgroup of order 2 or 4 is central in G, whence the result follows from Lemma 2.6. The final contradiction. We first show that P is maximal in G. Let L be any maximal subgroup of G that contains P. By induction, L = P O2 (L). Since O2 (G) L, we obtain [O2 (G), O2 (L)] ≤ O2 (G) ∩ O2 (L) = 1, hence O2 (L) ≤ CG (O2 (G)) ≤ O2 (G) by Lemma 2.8. Thus O2 (L) = 1 which implies that P is maximal in G. Moreover by Claim 2, O2 (G) is a maximal normal subgroup of G, ¯ and then G = G/O2 (G) is a simple group with a nilpotent maximal subgroup ¯ ¯= P/O2 (G). Assume that G is non-solvable. Then by Lemma 2.7, G ∼ L2 (q), where m ¯ be the maximal subgroup of L2 (q) q is a prime of the form 2 ± 1 ≥ 17. Let M which is isomorphic to the dihedral group D2s , where s > 1 is odd. Let M, K and ¯ A be the full inverse images of M , the 2-Sylow subgroup and the cyclic subgroup of ¯ in G. By Schur-Zassenhauss Theorem, A = O2 (G)T, where |T | = s. order s of M Also O2 (G) ≤ K ∈ Syl2 (M ) and M = KT, where A M. We next show that |D| ≤ |O2 (G)|. Assume false. Then |O2 (G)| < |D|. Now if |O2 (G)| < |D|/2 then ¯ ¯ ¯ G satisfies the hypothesis of Proposition 3.1 with |D| = |D|/|O2 (G)|, and hence G is 2-nilpotent, contradicts the simplicity of G. Thus we can assume that |O2 (G)| = ¯ |D|/2. Let H ≤ P be such that O2 (G) ≤ H and |H| = |H/O2 (G)| = 2. In this case, STRONGLY CLOSED 2-SUBGROUPS 5 P ≤ NG (H) < G and so NG (H) = P as P is maximal in G. By Lemma 2.4(b), we ¯ ¯ ¯ have NG (H) = P . Thus 1 = H is strongly closed in G and NG (H) is a 2-group. ¯ ¯ ¯ ¯ ¯ = P ∈ Syl2 (G) and so by Lemma 2.2, G is 2-nilpotent. This ¯ ¯ By Lemma 2.5, H contradiction shows that |D| ≤ |O2 (G)|. Therefore 2 < |D| ≤ |O2 (G)| < |K|, where K ∈ Syl2 (M ). By induction again, M = KT is 2-nilpotent and thus O2 (M ) = T M. Hence T ≤ CG (O2 (G)) ≤ O2 (G) and then T = 1, which contradicts the ¯ ¯ fact that |T | = s > 1. We conclude that G is solvable. Thus G must be a cyclic ¯ > 2 otherwise, G is a 2-group. Let r = |G| and ¯ subgroup of prime order. Clearly |G| R ∈ Sylr (G). Then G = O2 (G)R and r > 2, which implies that P = O2 (G) G, and hence G = NG (P ) is 2-nilpotent by Claim 3. The proof is now complete. Proof of Theorem 1.1. If P is cyclic or |D| > 2 or |D| = 2 but |P | > 2|D| = 4 then the theorem follows from Proposition 3.1. Thus we can assume that P is noncyclic, |D| = 2 and |P | = 4. It follows that every maximal subgroup of P is strongly closed in G, hence G is 2-nilpotent by Lemma 2.3. The proof is now complete. Proof of Theorem 1.3. By Theorem 1.2, G possesses a Sylow tower of supersolvable type. Let p be the largest prime divisor of |G|. If p = 2, then G must be a 2-group and hence it is supersolvable. Assume that p > 2. The proof now proceeds as in that of Theorem 3.5 in [7]. Proof of Theorem 1.4. By Lemma 2.4 and Theorem 1.3, E is supersolvable. Let p be the largest prime divisor of |E|. If p > 2, then the result follows as in Theorem 3.6 in [7]. Hence we can assume that p = 2 and so E is a 2-group. As G is supersolvable whenever G is a 2-group, we also assume that G is not a 2group. Since G/E is supersolvable, it has a Sylow tower of supersolvable type and so G/E is 2-nilpotent. Let K/E be the normal 2 -complement of G/E. By SchurZassenhauss Theorem, K = EA where A is of odd order. Let E ≤ P ∈ Syl2 (G). Then G = AP, where AE G. As |A| is odd, E ∈ Syl2 (AE) and AE satisfies the hypothesis of Theorem 1.1 so that AE is 2-nilpotent. Hence A = O2 (AE) AE G, and so A G. We have G/A ∼ P is supersolvable and by hypothesis, G/E is also = supersolvable. Since the class of supersolvable groups is a saturated formation, we have G/(A ∩ E) ∼ G is supersolvalbe. This completes the proof. = Acknowledgment. The author is grateful to the referee for his or her comments. The author is also grateful to Prof. Chris Parker for pointing out reference [2] and simplifying the proof of Lemma 2.11. References [1] M. Asaad, On p-nilpotence and supersolvability of finite groups, Comm. Algebra 34 (2006), no. 1, 189–195. [2] B. Baumann, Endliche nichtaufl¨sbare Gruppen mit einer nilpotenten maximalen Untero gruppe, J. Algebra 38 (1976), no. 1, 119–135. [3] M. Bianchi, A. Mauri, M. Herzog and L. Verardi, On finite solvable groups in which normality is a transitive relation, J. Group Theory 3 (2000), no. 2, 147–156. [4] D. Goldschmidt, 2-fusion in finite groups, Ann. of Math. (2) 99 (1974), no. 3, 70–117. [5] D. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. of Math. (2) 102 (1975), no. 3, 475–489. [6] Daniel Gorenstein, Finite Groups, Chelsea Publishing Company, Second Edition, 1980. [7] X. Guo and X. Wei, The influence of H-subgroups on the structure of finite groups, J. Group Theory 13 (2010), no. 2, 267–276. [8] B. Huppert, Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschaften, band 134, Springer-Verlag, Berlin-New York 1967. 6 HUNG P. TONG-VIET [9] M. Isaacs, Finite group theory, Graduate Studies in Mahtematics, 92 AMS, Providence, RI, 2008. Department of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK E-mail address: tongviet@maths.bham.ac.uk