On P-nipotence of Finite Groups

A subgroup H is said to be weakly c∗-normal in a group G if there exists a subnormal subgroup K of G such that HK = G and H ∩ K is s-quasinormally embedded in G. We give some results which generalize some authors’ results.


Introduction
In this paper the word group is always finite.Ore (1937, p150) gives quasinormality of subgroups.A subgroup H is said to be quasinormal in G if for every subgroup K of G such that HK = KH.A subgroup H of a group G is said to be s-quasinormal in G if H permutes with every Sylow subgroup of G.This concept was introduced by Kegel (1962, p 205),and extensively studied (Deskins, 1963, p126-131).Ballester-Bolinches and Pedraza-Aguilera (1998, p114) introduce the conception of s-quasinormally embedded in G if for each prime divisor p of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G. Wei and Wang (2007, p212)introduced the notion of c * -normality, a subgroup H of G is said to be c * -normal in G if there exists a subgroup K ≤ G such that G = HK and H ∩ K is s-quasinormally embedded in G.
For some notions and notations,the reader is referred to Robinson (1995)and Huppert (1968).

Some definitions and preliminary results
A subgroup H is called weakly c-normal in a group G if there exists a subnormal subgroup T of G such that G = HT and H ∩ T ≤ H G , where H G is the largest normal subgroup of G contained in H.The conception of weakly c-normality was introduced by Lu, Guo, and Shum (2002, p 5506).
Lemma 2.1 (Ballester-Bolinches and Pedraza-Aguilera, 1998, Lemma 1) Suppose that U is s-quasinormally embedded in a group G, and that H ≤ G and K ¡ G.
Lemma 2.2 Let G be a group.Then the following statements hold. (1 (2) If H is weakly c * -normal in G, then there exists a subnormal T of G such that G = HT and H ∩ T is s-quasinormally embedded in G. Then G/N = (H/N)(T N/N), where T N/N is subnormal in G/N and (H/N) ∩ (T N/N) is s-quasinormally embedded in G/N.Then H/N is weakly c * -normal in G/N.The converse part can be proved similarly.
(3) If H is weakly c * -normal in G, then there exists a subnormal subgroup T of G such that G = HT and (5) The result is obvious.
Lemma 2.3 Let M be a maximal subgroup of G and P a normal Sylow p-subgroup of G such that G = PM, where p is a prime, then P ∩ M is a normal subgroup of G.
Lemma 2.4 (Wei and Wang, 2007, Lemma 2.5) Let G be a group, K an s-quasinormal subgroup of G, P a Sylow p-subgroup of K where p is a prime divisor of |G|.
Lemma 2.5 (Li, Wang and Wei, 2003, Lemma 2.2) Let G be a group and P is s-quasinormal p-subgroup of G where p is a prime, then O P (G) ≤ N G (P).
(1) If N is normal in G of order p, then N is in Z(G).
(2) If G has cyclic Sylow p-subgroups, then G is p-nilpotent.
Lemma 2.7 (Huppert, 1968, IV, 5.4) Suppose that G is a group which is not p-nilpotent but whose proper subgroups are all p-nilpotent.Then G is a group which is not nilpotent but whose proper subgroups are all nilpotent.
Lemma 2.8 (Robinson, 1995, III, 5.2) Suppose that G is a group which is not nilpotent but whose proper subgroups are all nilpotent.Then (1) G has a normal Sylow p-subgroup for some prime p and G = PQ, where Q is a non-normal cyclic qsubgroup for some prime q p.
(5)If P is abelian, then exp(P) = p.Proof.Suppose that the result is false, then we chose a minimal order G as a counterexample.We will prove by the following steps: Steps 1.For every proper subgroup of G is p-nilpotent, thus G is a group which is not p-nilpotent but whose proper subgroups are all p-nilpotent.
Let M be a maximal subgroup of G, Then P ∩ M is a maximal p-subgroup of P. By hypothesis, P ∩ M is weakly c * -normal in G and so P ∩ M is weakly c * -normal in M by lemma 2.2(1).Thus M, P ∩ M satisfies the hypotheses of the theorem, the minimal choice of G implies that M is p-nilpotent.Then we have that G is not p-nilpotent but all proper subgroups are p-nilpotent.Then, by lemma 2.7 and lemma 2.8(1), G has a normal Sylow p-subgroup for some prime p and G = PQ, where Q is a non-normal cyclic q-subgroup for some prime q p.
Steps 2. Let L be a minimal normal subgroup of G contained in P, then G/L is p-nilpotent, L is the unique minimal normal of G and L Φ(G).
Since P/L is a Sylow p-subgroup of G/L, we have M/L is a maximal subgroup in P/L, where M is a maximal subgroup of P. Since M is weakly c * -normal in G, by lemma 2.2(2) M/L is weakly c * -normal in G/L.Thus G/L, P/L satisfies the hypotheses of the theorem and so we have G/L is p-nilpotent by the minimal choice of G.If L 1 is an another minimal normal subgroup, then G/1 G/L × G/L 1 is p-nilpotent and so L is unique.If L ≤ Φ(G), then G/Φ(G) is p-nilpotent, and so is G, a contradiction.
If Φ(P) = 1, then P is abelian.By steps 1 and lemma 2.8(5), exp(P) = p.If |P/Φ(P)| = P n and P/Φ(P) =< and < x i > char P, where i is nature number.And since P is normal in G, then < x i > are normal p-subgroup of G of order p.Thus by lemma 2.6(1), we have < x i >≤ Z(G) for all i = 1, 2, • • • , n, then P ≤ Z(G), then G is p-nilpotent, a contradiction.Thus Φ(P) 1.
Steps 4. L is a Sylow p-subgroup of G.
By steps 3, Φ(P) 1, then L ≤ P. If L < P, then for a maximal subgroup M of P, M is weakly c * -normal in G and so there exists a subnormal subgroup K such that G = MK and M ∩ K is s-quasinormally embedded in G.We consider the following cases.
Let S = M ∩ K. Then S is s?quasinormally embedded in G. Thus there exists an s-quasinormal subgroup R such that S is a Sylow p-subgroup of R. Then by lemma 2.5, we have O p (G) ≤ N G (S ) and so S is normal in G, then we have, S = P or S = L.If S = P. On the other hand, |M| < |P|, a contradiction.Then S = L is a minimal normal Sylow p-subgroup of some s-quasinormal subgroup R of G, then for any sylow q-subgroup Q, we have RQ = QR is a subgroup of G and, if QS < G, Q ¡ RQ by (1), and so LQ = L × Q.By steps 1 and Burnside's theorem, we have G is solvable.Thus Q ≤ C G (L) ≤ L, a contradiction.Then QS = G, then ¢ www.ccsenet.org/jmrVol. 1, No. 1 ISSN: 1916-9795 G = PQ = QS and so P = S g for some g ∈ Q, a contradiction.
By steps 4, L = P is a Sylow p-subgroup of G, then, by hypothesis, maximal subgroup M of L = P is weakly -normal in G. Then by lemma 2.10 there exists a subnormal subgroup K of G such that G = MK and M ∩ K ≤ M sG .Since M sG < L = P < L = P, then M sG = 1 and LQ/L is p-nilpotent since G/L is p-nilpotent by steps 2, where Q is a Hall p'-subgroup of G, then LQ/L ¡ G/L and so LQ ¡ G.
Corollary 3.1 (Wei and Wang, 2007, Theorem 3.1) Let G be a group, P a Sylow p-subgroup of G, where p is a prime divisor of |G| with (|G|, p − 1) = 1.If all maximal subgroups of P are c * -normal in G, then G is p-nilpotent.
Theorem 3.2 Let G be a group, P a Sylow p-subgroup of G, where p is a prime divisor of |G| with (|G|, p−1) = 1.If all cyclic subgroups of P of order p or 4 (if p = 2) are weakly c * -normal in G, then G is p-nilpotent.
Proof.Suppose that the result is false, then we chose a minimal order G as a counterexample.We will prove by the following steps: Steps 1.Let M be a proper subgroup of G, then M is p-nilpotent.So G is not p-nilpotent but all proper subgroups are p-nilpotent.Thus G = PQ, where P is a normal Sylow p-subgroup of G and Q is a non-normal cyclic Sylow q-subgroup of G.And so by Burnside's theorem G is solvable.Then M ∩ P is a Sylow p-subgroup of M. By hypothesis, for every cyclic subgroup of P of order p or 4 (if p = 2) is weakly -normal in G, then By lemma 2.2(1), for every cyclic subgroup of P M of order p or 4 (if p = 2) is weakly c * -normal in M. Then M, M ∩ P satisfies the hypotheses of the theorem, M is p-nilpotent by the minimal choice of G, so we have: G is not p-nilpotent but all proper subgroups are p-nilpotent and so by lemma 2.7 and lemma 2.8(1), G = PQ, where P is a normal Sylow p-subgroup of G and Q is a non-normal cyclic Sylow q-subgroup of G.
Steps 2. Let L be a minimal normal subgroup of G contained in P, then L is unique minimal normal p-subgroup for some prime of |G|, G/L is p-nilpotent and L Φ(G).Furthermore, Since all cyclic subgroups of P of order p or 4(if p = 2) is weakly c * -normal in G, then by lemma 2.2(2) all cyclic subgroups of P/L with order p or 4 (if p = 2) is weakly c * -normal in G/L, then the minimal choice of G implies that G/L is p-nilpotent.If L ≤ Φ(G), then G/Φ(G) is p-nilpotent and G is p-nilpotent, a contradiction.By lemma 2.6 (Li, etc, 2003), F(G) = L.By steps 1, solubility of G implies that L ≤ C G (F(G)) ≤ F(G) and so C G (L) = F(G) = L as L is abelian.
By steps 2 C G (L) = F(G) = L.But on the other hand, for x ∈ P, < x > is weakly c * -normal in G, then there exists a subnormal subgroup T of G such that G =< x > T and < x > ∩T is s-quasinormally embedded in G.By lemma 2.7 and lemma 2.8, we have if p is odd or P is abelian, then exp(P) = p or if p = 2exp(P) = 4. Since F(G) =< x 1 , x 2 , • • • , x n >= L, | < x i > | = p or 4 and < x i > char P since P is normal in G. Thus Corollary 3.2 (Li and Wang, 2004, Theorem 4.1) Suppose G is a group, p is a fixed prime number.If every element of P p (G) is contained in Z ∞ (G).If p = 2, every cyclic subgroup of order 4 of G is s-quasinormal in G, then G is p-nilpotent.
Lemma 2.9 Let H be a subgroup of G. Then H is weakly c * -normal in G if and only if there exists a subgroup K such that G = HK and H ∩ K = H sG .Proof.⇐ It is clear.⇒By definition 2.1, there exists a subnormal subgroup L of G such that G = HL andH ∩ L ≤ H sG If H ∩ L < H sG , note that K = LH sG , then HK = HLH sG = LHH S G = LH = G and hence H ∩ K = H ∩ LH sG = (H ∩ L)H sG = H sG .¢ www.ccsenet.org/jmrJournal of Mathematics ResearchMarch, 20093.Main resultsTheorem 3.1 Let G be a group, P a Sylow p-subgroup of G, where p is a prime divisor of |G| with (|G|, p−1) = 1.If all maximal subgroups of P are weakly c * -normal in G, then G is p-nilpotent.