Proc. Indian Acad. Sci. (Math. Sci.) Vol. 125, No. 2, May 2015, pp. 173–179. c Indian Academy of Sciences On p-supersolvability of finite groups IZABELA AGATA MALINOWSKA Institute of Mathematics, University of Białystok, 15-245 Białystok, Ciołkowskiego 1 M, Poland E-mail: [email protected] MS received 4 April 2013; revised 3 June 2014 Abstract. A number of authors have studied the structure of a group G under the assumption that some subgroups of G are well located in G. We will obtain some new criteria of p-supersolvability and p-nilpotency of groups. Keywords. p-Supersolvability; p-nilpotency; weakly τ-embedded subgroups. 2010 Mathematics Subject Classification 20D10, 20D20. 1. Introduction Throughout the paper, all groups are finite. Recall that a subgroup H of a group G is said to permute with a subgroup K if HK = KH. A subgroup H is said to be an s-permutable subgroup of a group G if H permutes with all Sylow subgroups of G.Guoet al.[7], Lukyanenko and Skiba [9, 10] introduced the following concepts which generalize s- permutability. A subgroup H of G is said to be s-embedded in G if G has a normal subgroup T such that HT is s-permutable in G and H ∩ T HsG, where HsG denotes the subgroup generated by all those subgroups of H which are s-permutable in G [7]. A subgroup H of G is said to be τ-permutable (τ-quasinormal)inG if H permutes with all Sylow q-subgroups Q of G such that (q, |H|) = 1 and (|H|, |QG|) = 1 [9, 10]. More recently, Chen and Guo [4] introduced the concept, which covers both τ-permutability and s-embedded property. A subgroup H of G is said to be weakly τ-embedded in G if there exists a normal subgroup T of G such that HT is s-permutable in G and H ∩ T HτG, where HτG is the subgroup generated by all those subgroups of H which are τ-permutable in G. In [4], Chen and Guo obtained some results on p-nilpotency and supersolvability by using the notion of weakly τ-embedding. In this paper, we continue to investigate this concept and we give some criteria of p-supersolvability. Some consequences of the main results are also derived. We will need the following lemmas. Lemma 1.1 (Lemma 2.10 of [4]). Assume that H is weakly τ-embedded in G, U G, N G and p is a prime. (1) If H U, then H is weakly τ-embedded in U. (2) If H is a p-group and N H, then H/N is weakly τ-embedded in G/N. (3) If (|H |, |N|) = 1, then HN/N is weakly τ-embedded in G/N. 173 174 Izabela Agata Malinowska Lemma 1.2 (Lemma 2.3 of [10] and Lemma 2.7 of [4]). Assume that H U G, N G and p is a prime. Then (1) If H is a p-group, then HτG is τ-permutable in G and HG HτG; (2) HτG HτU; (3) If H is a p-group, then HτGN/N (HN/N)τ(G/N); (4) If (|H |, |N|) = 1, then HτGN/N (HN/N)τ(G/N). Lemma 1.3 (Lemma 2.2 of [10] and Lemma 2.6 of [4]). Let G be a group and H K G. (1) If H is τ-permutable in G, then H is τ-permutable in K; (2) Assume that N is normal in G and π(KN/N) = π(K).IfK is τ-permutable in G, then KN/N is τ-permutable in G/N. (3) Assume that H is normal in G. Then EH/H is τ-permutable in G/H for every τ- permutable subgroup E in G satisfying (|H|, |E|) = 1. (4) If H is τ-permutable in G and H Op(G) for some prime p, then H is s-permutable in G. Lemma 1.4. Let P be a normal p-subgroup of a group G. (1) Assume that P is an elementary abelian. If all maximal subgroups of P are τ- permutable in G, then each minimal normal subgroup of G contained in P is of order p; (2) Assume that P/(P)is a minimal normal subgroup of G/(P ) and H P .IfH is weakly τ-embedded in G, then H(P )/(P ) is s-permutable in G/(P ). Proof. Follow by Lemma 1.3(4) and Lemma 1.5 of [11]. Lemma 1.5 (VI.4.10 of [8]). Let A and B be subgroups of G such that G = AB.If AgB = BAg for all g ∈ G, then either A or B is contained in a proper normal subgroup of G. Lemma 1.6. Let p be a prime and P be a Sylow p-subgroup of a nonabelian simple group G. Then no nontrivial subgroup of P is weakly τ-embedded in G. Proof. Assume that there exists a nontrivial subgroup H of P such that H is weakly τ- embedded in G. By definition, there exists a normal subgroup T of G such that HT is s-permutable in G and H ∩ T HτG. Since G is a nonabelian simple group, H is τ- permutable in G by Lemma 1.2(1). Hence HSq G for any Sylow q-subgroup Sq of G, where q is a prime, q = p. By Lemma 1.5 H or Sq is contained in a proper normal subgroup of G. Since G is a nonabelian simple group, we get a contradiction. Lemma 1.7. Let G be a group and let H be a p-subgroup of G for some p ∈ π(G). Assume that H is τ-permutable in G.IfN is a normal subgroup of G, then H ∩ N is also τ-permutable in G. Proof. Let Q be a Sylow q-subgroup of G, where q is a prime such that q = p and G (p, |Q |) = 1. Since |N|q =|HN|q and N ∩ Q is a Sylow q-subgroup of N,wehave On p-supersolvability of finite groups 175 that N ∩ Q = HN ∩ Q. Hence (H ∩ Q)(N ∩ Q) = HN ∩ Q. By A.1.2 of [5], we have HQ∩ NQ = (H ∩ N)Q. Thus H ∩ N is τ-permutable in G. Lemma 1.8. Let G be a group, H a subgroup of G and let N be the unique minimal normal subgroup of G. Assume that H is weakly τ-embedded in G.IfH is a p-group for some prime p, then H ∩ N is τ-permutable in G. Proof. By definition, there exists a normal subgroup T of G such that HT is s-permutable in G and H ∩ T HτG. Then H ∩ N H ∩ T HτG. Hence HτG ∩ N H ∩ N HτG ∩ N. Therefore H ∩ N = HτG ∩ N is τ-permutable in G by Lemmas 1.2(1) and 1.7. 2. Main results Theorem 2.1. Let p be a prime dividing the order of a group G. Assume that all max- imal subgroups of every Sylow p-subgroup of G are weakly τ-embedded in G. Then either G is a group whose Sylow p-subgroups are of order p or G is a p-supersolvable group. Proof. Let P be a Sylow p-subgroup of G. Assume that the result is false and G is a counterexample of minimal order. We make the following claims: (1) Op′ (G) = 1: This follows directly by Lemma 1.1(3). (2) If M is a proper subgroup of G and p does not divide |G : M|, then either M is p- supersolvable or M is a non-p-solvable group whose Sylow p-subgroups have order p:IfS is a Sylow p-subgroup of M, then S is a Sylow p-subgroup of G. Hence by Lemma 1.1(1) all maximal subgroups of every Sylow p-subgroup of M are weakly τ-embedded in M. By the minimality of G, (2) follows. (3) Let N be a minimal normal subgroup of G. Then either G/N is p-supersolvable or G/N is a non-p-solvable group whose Sylow p-subgroups are of order p:We will show that G/N satisfies the assumptions of the theorem. Let M/N be a max- imal subgroup of PN/N. Then M = P1N for some maximal subgroup P1 of P . It follows that P1 ∩ N = P ∩ N is a Sylow p-subgroup of N. By the assump- tions, there exists a normal subgroup T of G such that P1T is s-permutable in G and P1 ∩ T (P1)τG. Then P1TN/N is s-permutable in G/N and TN/N is nor- mal in G/N by Lemma 1.2.7 of [1]. If N T , then (P1 ∩ N)(T ∩ N) = N = N ∩ P1T .IfN ∩ T = 1, then (P1 ∩ N)(T ∩ N) = P1 ∩ N = N ∩ P1T , since N and T are normal subgroups of G. By A.1.2 of [5], NP1 ∩ NT = N(P1 ∩ T). Hence (P1N/N) ∩ (T N/N) = (P1 ∩ T)N/N (P1)τGN/N (P1N/N)τ(G/N) by Lemma 1.2(3). Hence M/N is weakly τ-embedded in G/N. Therefore G/N satisfies the assumptions of the theorem. By minimality of G, either G/N is p- supersolvable or G/N is a non-p-solvable group whose Sylow p-subgroups are of order p. (4) Let N be a minimal normal subgroup of G and denote Np = N ∩P .