1 Introduction

All groups considered in this paper are finite. We use conventional notions and notation, as in [5]. G always denotes a finite group, |G| is the order of G, \(\pi (G)\) denotes the set of all primes dividing |G|, and \(G_p\) is a Sylow p-subgroup of G for a prime \(p\in \pi (G)\).

In [13], Wang defined the c-normality of a subgroup as follows and prove that a finite group G is solvable if and only if every maximal subgroup of G is c-normal in G.

Definition 1.1

([13, Definition 1.1]) Let H be a subgroup of a finite group G. We say that H is c-normal in G if there exists a normal subgroup \(N\) of G such that \(HN=G\) and \(H\cap N\le H_{G}\).

The basic properties of c-normality are as follows.

Lemma 1.2

(see [13, Lemma 2.1] and [9, Lemma 2.4]) Let G be a group. Then

  1. (1)

    If H is normal in G, then H is c-normal in G.

  2. (2)

    If H is c-normal in G and \(H\le K\le G\), then H is c-normal in K.

  3. (3)

    Suppose that K is a normal subgroup of G and that H is c-normal in G. Then HK / K is c-normal in G / K when \(K\le H\) or \((|H|,|K|)=1\).

Let G be a finite group. Several authors successfully use the c-normal property of some subgroups of G of prime power order to determine the structure of G (see [1, 2, 8–12]). Many results in these previous papers have the following form: Let E be a normal subgroup of G and \(\mathcal F\) be a saturated formation containing the class of all supersolvable groups. Suppose that G / E is in \(\mathcal F\). If for each prime divisor p of |E|, some p-subgroups of E are c-normal G, then \(G\in \mathcal F\). Actually, in a more general case, if we can get a criterion that E lies in the \(\mathcal F\)-hypercenter, then \(G/E \in \mathcal F\) implies that \(G \in \mathcal F\). In order to get good results, many authors have to impose the c-normal hypotheses on all the prime divisors or the minimal or maximal divisor p of |G| rather than any prime divisor. In this paper, we try to get more general results.

Now let p be a fixed prime. In this paper, we focus on how a normal subgroup E embedded in G provided every \(p\)-subgroup of E with some fix order is c-normal in G. For this purpose, we introduce the concept of p-hypercentrally embedded:

Definition 1.3

Let G be a finite group. A normal subgroup E of G is said to be \(p\)-hypercentrally embedded in G if every p-chief factor of G below E is cyclic.

The main result of this paper is the following theorem.

Theorem A

Let p be a fixed prime and E be a normal subgroup of a finite group G. Suppose that \(E_{p}\) is a Sylow \(p\)-subgroup of E and \(p^{d}\) is a prime power such that \(1<p^{d}\le \max (|E_{p}|/p,p)\). If all the subgroups of \(E_{p}\) with order \(p^{d}\) and \(2p^{d}\) (if a quaternion group is involved in \(E_{p}\)) are c-normal in G, then E is \(p\)-hypercyclically embedded in G.

2 Proof of the Results

First, we need some results on c-supplemented subgroups of finite groups. Following [3], a group H is said to be c-supplemented in G if there exists a subgroup K of G such that \(G=HK\) and \(H\cap K\le H_{G}\). It is clear from the definition that if a subgroup H of G is c-normal in G, then H is c-supplemented in G.

Lemma 2.1

If \(N\) is a minimal abelian normal subgroup of G, then any proper subgroup of \(N\) is not c-supplemented in G.

Proof

Suppose that this Lemma is not true and let \(H\) be a proper subgroup of \(N\) such that H is c-supplemented in G. Obviously, \(H_{G}=1\) since \(H_{G}<N\) and \(N\) is a minimal normal subgroup of G. By the definition of c-supplemented subgroups, there exists a subgroup \(M\) of G such that \(G=HM\) with \(H\cap M\le H_{G}=1\). Hence \(NM\ge HM=G\). Since \(N\) is abelian, we know that \(N\cap M\unlhd G\). Hence \(N\cap M=1\). Therefore, we have \(|G|=|NM|=|N||M|>|H||M|=|HM|\), a contradiction. \(\square \)

For a saturated formation \(\mathcal F\), the \(\mathcal F\)-hypercenter of a group G is denoted by \(Z_{\mathcal {F}}(G)\) (see [5, p. 389, Notation and Definitions 6.8(b)]). Let \(\mathcal U\) denote the class of all supersolvable groups and let \(\mathcal N\) denote the class of all nilpotent groups. Suppose that A is a normal subgroup of G. It is clear that \(A\le Z_\mathcal{U}(G)\) if and only if every chief factor of G below A is cyclic and that \(A\le Z_\mathcal{N}(G)\) if and only if every chief factor of G below A is central. In [1], Asaad gave the following result: Let P be a nontrivial normal \(p\)-subgroup, where p is an odd prime. If every minimal subgroup of P is c-supplemented in G, then \(P\le Z_{\mathcal U}(G)\). It is helpful to give a result for \(p=2\). In fact, we have the following proposition:

Proposition 2.2

Let P be a normal 2-subgroup of G. If all minimal subgroups of P and all cyclic subgroups of P with order 4 (if a quaternion group is involved in P) are c-supplemented in G, then \(P\le Z_\mathcal{N}(G)\).

Proof

Let Q be a Sylow q-subgroup of G, where q is prime different from p. We claim that PQ is nilpotent. Suppose that the claim is not true and let H be a minimal non-nilpotent subgroup of PQ. Then \(H=[H_{2}]H_{q}\), where \(H_{q}\in Syl_{q}(H)\) and \(H_2\) is a normal Sylow 2-subgroup of H. By Itô’s result (see [4, Chap. 3, 5.2]), we have that \({{\mathrm{exp}}}(H_{2})\le 4\) and that \(H_{q}\) acts irreducibly on \(H_{2}/\Phi (H_{2})\). It is easy to see that \(|H_{2}/\Phi (H_{2})|\ge 4\). Clearly, \(H_q\) acts nontrivially on \(H_2\) but acts trivially on any proper \(H_q\)-invariant subgroup of \(H_2\). It follows by the reduction theorem of Hall and Higmman (see [7, Chap. 5 Theorem 3.7]) that \(H_{2}'=\Phi (H_{2})\).

Case 1. Suppose first that a quaternion group is involved in \(H_2\). Then \({{\mathrm{exp}}}(H_{2})=4\), and we may take a subgroup \(\langle x\rangle \le H_2\) of G of order 4 and \(\langle x\rangle \nleq \Phi (H_2)\). By hypotheses \(\langle x\rangle \) is c-supplemented in G, and thus \(\langle x\rangle \Phi (H_{2})/\Phi (H_{2})\ne 1\) is c-supplemented in \(H/\Phi (H)\), which contradicts Lemma 2.1.

Case 2. Suppose that \(H_2\) is quaternion free. Assume that \(H_{2}\) is abelian. Then \(H'_{2}=\Phi (H_2)=1\) and thus \(H_2\) is a minimal normal subgroup of H. It follows from Lemma 2.1 that \(H_2=1\), a contradiction. Assume that \(H_2\) is not abelian. Applying [6, Theorem 2.7], \(H_{q}\) acts on \(H_{2}/\Phi (H_{2})\) with at least one fixed point. This implies that \(|H_{2}/\Phi (H_{2})|=2\), a contradiction.

The above proof shows that PQ is nilpotent as claimed. In particular, P centralizes all odd elements of G. Thus for any G-chief factor H / K of P, \(G/C_{G}(H/K)\) is a 2-group. By [5, A, Lemma 13.6], we have \({{\mathrm{O}}}_{2}(G/C_{G}(H/K))=1\). It follows that \(G/C_{G}(H/K)=1\) and thus \(P\le Z_\mathcal{N}(G)\). \(\square \)

As an application of Proposition 2.2, we have

Corollary 2.3

If all minimal subgroups of \(G_{2}\) and all cyclic subgroups of \(G_{2}\) with order 4 (if a quaternion group is involved in \(G_{2}\)) are c-supplemented in G, then G is 2-nilpotent.

Proof

Suppose that this corollary is not true and let G be a counterexample with minimal order. Obviously, the hypotheses are inhered by all subgroups of G, hence G is a minimal non- 2-nilpotent group. It follows that \(G_{2}\) is a normal subgroup in G. Applying Proposition 2.2 to \(G_{2}\), we get a contradiction. \(\square \)

By combining [1, Theorem 1.1] and Proposition 2.2, we have

Lemma 2.4

Let P be a normal \(p\)-subgroup of G. If all cyclic subgroups of P with order p or 4 (if a quaternion group is involved in P) are c-supplemented in G, then \(P\le Z_{\mathcal U}(G)\).

Next, we will show that if some class of \(p\)-subgroups of G is c-normal in G, then G is \(p\)-solvable.

Lemma 2.5

If \(G_{p}\) is c-normal in G then G is \(p\)-solvable.

Proof

Suppose that this Lemma is not true and consider G to be a counterexample with minimal order. By Lemma 1.2 (3), the hypothesis holds for both \(G/{{\mathrm{O}}}_{p}(G)\) and \(G/{{\mathrm{O}}}_{p'}(G)\), thus the minimal choice of G implies that \({{\mathrm{O}}}_{p}(G)={{\mathrm{O}}}_{p'}(G)=1\). By the definition of c-normality, there exists a normal subgroup H of G such that \(G=G_{p}H\) and \(H\cap G_{p}\le (G_{p})_{G}\). But \((G_{p})_{G}={{\mathrm{O}}}_{p}(G)=1\), hence H is a normal \(p'\)-subgroup of G. The fact that \({{\mathrm{O}}}_{p'}(G)=1\) then indicates that \(H=1\) and thus \(G=G_{p}\), a contradiction. \(\square \)

Lemma 2.6

Let G be a finite group and \(p^d\) be a prime power such that \(3\le p^{d}\le |G|_{p}\). If all subgroups of G with order \(p^d\) are c-normal in G, then G is \(p\)-solvable.

Proof

By Lemma 2.4, we may assume that \(p^{d}<|G|_p\). Let D be any subgroup of order \(p^{d}\). By the hypotheses, there exists a normal subgroup H of G such that \(G = DH\) and \(D \bigcap H\le D_{G}\). Assume that \(H < G\). Since G / H is a p-group, we may take a normal subgroup M of G such that \(M \ge H\) and \(|G : M | = p\). As \(p^{d}<|G|_p\), \(p^{d}\le |M|_{p}\).

Clearly, all subgroups of M with order \(p^{d}\) are c-normal in M. It follows by induction that M is p-solvable, and so is G. Hence we may assume that \(H = G\) and then \(D = D_{G}\) is normal in G. Assume that D is not a minimal normal subgroup of G. Let V be a minimal G-invariant subgroup of D and \(|V|=p^{e}\). Then \(p \le |V | < p^{d}\) and all subgroups of G / V with order \(p^{d-e}\) are c-normal in G / V. By Induction G / V is p-solvable, and so is G. Hence we may assume that D is minimal normal in G whenever D is a subgroup of order \(p^{d}\).

Note that if all subgroups of order \(p^d\) are contained in Z(G), then G is \(p\)-nilpotent by a well-known result of Itô (see [4, IV, 5.3]). Hence we may assume that there is a subgroup U of order \(p^{d}\) such that \(U\nleq Z(G)\). Suppose that \(|U| = p^{d}\ge p^{2}\). Let K be a subgroup of order \(p^{d+1}\) such that \(U<K\). Clearly U is not cyclic, and hence there is a maximal subgroup \(U_1\) of K such that \(U_1\ne U\). Since \(U_1\) is normal as assumed, we get that \(U\cap U_{1}\) is a nontrivial G-invariant subgroup of U, and this contradicts the minimal normality of U. Hence \(|U|=p\). Observe that \(G/C_{G}(U)\) is a \(p'\)-group and that \(C_{G}(U)<G\). It follows by induction that \(C_{G}(U)\) is p-solvable, and so is G. \(\square \)

Now, we will study the properties of \(p\)-hypercyclically embedding. Clearly, if a normal subgroup E is p-hypercyclically embedded in G, then E is \(p\)-solvable and every normal subgroup of G contained in E is also p-hypercyclically embedded in G. The following lemma shows that for a \(p\)-solvable normal subgroup E, we can deduce that E is p-hypercyclically embedded in G if the maximal p-nilpotent normal subgroup of E (denoted by \(F_{p}(E)\)) is p-hypercyclically embedded in G.

Lemma 2.7

A \(p\)-solvable normal subgroup E is p-hypercyclically embedded in G if and only if \(F_{p}(E)\) is p-hypercyclically embedded in G.

Proof

We only need to prove the sufficiency. Suppose that the assertion is false and let (G, E) be a counterexample with \(|G|+|E|\) minimal. We claim that \({{\mathrm{O}}}_{p'}(E)=1\). Indeed, since \(F_{p}(E/{{\mathrm{O}}}_{p'}(E))=F_{p}(E)/{{\mathrm{O}}}_{p'}(E)\), it is easy to verify that the hypotheses hold for \((G/{{\mathrm{O}}}_{p'}(E), E/{{\mathrm{O}}}_{p'}(E))\). If \({{\mathrm{O}}}_{p'}(E)\ne 1\), then the minimal choice of (G, E) implies that \(E/{{\mathrm{O}}}_{p'}(E)\) is p-hypercyclically embedded in \(G/{{\mathrm{O}}}_{p'}(E)\). Clearly \({{\mathrm{O}}}_{p'}(E)\) is p-hypercyclically embedded in G. Therefore, E is p-hypercyclically embedded in G, a contradiction.

Let \(N\) be a minimal normal subgroup of G contained in E. \(N\) is an abelian normal \(p\)-subgroup since E is \(p\)-solvable and \({{\mathrm{O}}}_{p'}(E)=1\). Consider the group \(C_{E}(N)/N\). Let \(L/N={{\mathrm{O}}}_{p'}(C_{E}(N)/N)\) and \(K\) be a Hall \(p'\)-subgroup of L. Then \(L=KN\). Since \(K\le L\le C_{E}(N)\), we have \(K={{\mathrm{O}}}_{p'}(L)\le {{\mathrm{O}}}_{p'}(G)=1\). Consequently, \({{\mathrm{O}}}_{p'}(C_{E}(N)/N)=1\) and we have \(F_{p}(C_{E}(N)/N)={{\mathrm{O}}}_{p}(C_{E}(N)/N)\le {{\mathrm{O}}}_{p}(E)/N=F_{p}(E)/N\). As a result, we know that the hypotheses hold for \((G/N, C_{E}(N)/N)\) and the minimal choice of (G, E) yields that \(C_{E}(N)/N\) is p-hypercyclically embedded in G / N. But \(N\le F_{p}(G)\) and thus \(N\) is also p-hypercyclically embedded in G. It follows that \(C_{E}(N)\) is p-hypercyclically embedded in G.

Since N is a normal \(p\)-subgroup that is p-hypercyclically embedded in G, \(|N|=p\). It yields that \(G/C_{G}(N)\) is a cyclic group. As a result, \(EC_{G}(N)/C_{G}(N)\) is \(p\)-hypercyclically embedded in \(G/C_{G}(N)\). Note that \(E/C_{E}(N)=E/(E\cap C_{G}(N))\) is G-isomorphic to \(EC_{G}(N)/C_{G}(N)\) and \(E/C_{E}(N)\) is \(p\)-hypercyclically embedded in \(G/C_{E}(N)\). But \(C_{E}(N)\) is p-hypercyclically embedded in G and thus E is or p-hypercyclically embedded in G, a final contradiction. \(\square \)

Denote \(\mathcal {A}(p-1)\) as the formation of all abelian groups of exponent divisible by \(p-1\). The following proposition is well known:

Lemma 2.8

([15, Theorem 1.4]) Let H / K be a chief factor of G and p be a prime divisor of |H / K|. Then \(|H/K|=p\) if and only if \(G/C_{G}(H/K)\in \mathcal {A}(p-1)\).

Let f be a formation function and E be a normal subgroup of G. We say that G acts f-centrally on E if \(G/C_{G}(H/K)\in f(p)\) for every chief factor H / K of G below E and every prime p dividing |H / K| ([5], p. 387, Definitions 6.2). Fixing a prime p, define a formation function \(g_p\) as follows:

$$\begin{aligned} g_{p}(q)= {\left\{ \begin{array}{ll} \mathcal {A}(p-1)&{}\text {(if}\,q=p\text {)}\\ \text {all finite group}&{}\text {(if}\, q\ne p\text {)} \end{array}\right. } \end{aligned}$$

From Lemma 2.8, we can see that E is \(p\)-hypercyclically embedded in G if and only if G acts \(g_{p}\)-centrally on E. By applying [5, p. 388, Theorem 6. 7], we get the following useful results:

Lemma 2.9

A normal subgroup E of G is \(p\)-hypercyclically embedded in G if and only if \(E/\Phi (E)\) is \(p\)-hypercyclically embedded in \(G/\Phi (E)\).

Then the following lemma is evident.

Lemma 2.10

Let \(K\) and \(L\) be two normal subgroups of G contained in E. If E / K is \(p\)-hypercyclically embedded in G / K and E / L is \(p\)-hypercyclically embedded in G / L, then \(E/(L\cap K)\) is \(p\)-hypercyclically embedded in \(G/(L\cap K)\).

The following proposition indicates that Theorem A holds when \(p^d=p\).

Proposition 2.11

Let E be a normal subgroup of G. If all cyclic subgroups of \(E_{p}\) with order p and 4 (if a quaternion group is involved in \(E_{p}\)) are c-normal in G, then E is \(p\)-hypercyclically embedded in G.

Proof

Note that E is \(p\)-solvable by Lemma 2.6. Suppose that \({{\mathrm{O}}}_{p'}(E)>1\). Since the hypotheses hold for \(G/{{\mathrm{O}}}_{p'}(E)\), we conclude by induction that \(E/{{\mathrm{O}}}_{p'}(E)\) is \(p\)-hypercyclically embedded in \(G/{{\mathrm{O}}}_{p'}(E)\) and thus E is \(p\)-hypercyclically embedded in G. Suppose that \({{\mathrm{O}}}_{p'}(E)=1\). By Lemma 2.4, \({{\mathrm{O}}}_{p}(E)\le Z_{\mathcal U}(G)\). As \({{\mathrm{O}}}_{p'}(E)=1\), \(F_{p}(E)={{\mathrm{O}}}_{p}(E)\). It follows that E is \(p\)-hypercyclically embedded in G by Lemma 2.7. \(\square \)

With the aid of the preceding results, we can now prove Theorem A.

Proof of Theorem A

Suppose that Theorem A is not true and let (G, E) be a counterexample such that \(|G|+|E|\) is minimal. Then the minimal choice of (G, E) implies that \({{\mathrm{O}}}_{p'}(E)=1\). If \(|E_{p}|=p\), then \(E_{p}\) itself is c-normal in G and by Lemma 1.2, \(E_{p}\) is also c-normal in E. By Lemma 2.5, we know that E is \(p\)-solvable and consequently E is \(p\)-hypercyclically embedded in G since \(|E_{p}|=p\). Therefore, we may assume that \(|E_{p}|>p\) and \(1<p^{d}<|E_{p}|\). By Proposition 2.11, we may further assume that \(p^{d}>p\). By Lemma 2.6, E is \(p\)-solvable. We derive a contradiction through the following steps.

(1)  If N is a minimal G-invariant subgroup of E, then \(|N|>p\).

Suppose that \(|N|=p\), then \(p^{d}>|N|\) by the assumption that \(p^{d}>p\). Hence (G / N, E / N) also satisfies the hypotheses of this Theorem and E / N is \(p\)-hypercyclically embedded in G / N by the choice of (G, E). Since \(|N|=p\), E is \(p\)-hypercyclically embedded in G, a contradiction. Hence \(|N|>p\).

(2)  If N is a minimal G-invariant subgroup of E, then \(p^{d}>|N|\).

By Lemma 2.1 we have \(p^{d}\ge |N|\). Suppose that \(p^{d}=|N|\). Since \(p^{d}<|E_{p}|\) by our assumption, \(E_p\) has a subgroup H such that N is a maximal subgroup of H. By (1), \(N\) is not cyclic and so is H. Hence we can choose a maximal subgroup \(K\) of \(H\) other than N. Obviously we have \(H=NK\). If \(N\cap K=1\), then \(|N|=|H|/|K|=p\), a contradiction. Thus \(N\cap K\ne 1\) and \(|K:K\cap N|=|KN:N|=|H:N|=p\). Since \(K_{G}\cap N\le K\cap N<N\), we have \(K_{G}\cap N=1\). Assume that \(K_{G}> 1\). Then \(|K_{G}|=|K_{G}N/N|=|H/N|=p\) and this contradicts (1). Therefore we have \(K_{G}=1\). Since \(|K|=|N|=p^{d}\), \(K\) is c-normal in G by the hypotheses of this theorem. So there exists a proper normal subgroup \(L\) of G such that \(G=KL\) and \(K\cap L\le K_{G}=1\). Since \(K\cap N\ne 1\) and \(K\cap L=1\), we have \(N\ne L\) and thus \(N\cap L=1\). Consequently, \(|NL|=|N||L|=|K||L|=|KL|=|G|\) and thus \(G=NL\). Let \(M\) be a maximal subgroup of G containing L, then \(|G:M|=p\) since G / L is a \(p\)-group. Obviously \(G=NM\) and \(N\cap M=1\). But then \(|N|=|G:M|=p\), a contradiction.

(3)  \(\Phi (E)=1\) and E contains a unique minimal G-invariant subgroup, say N.

Let L be any minimal G-invariant subgroup of E. Since \(p^{d}>|L|\) by (2), it is easy to verify that (G / L, E / L) satisfies the hypotheses of this theorem. Thus the minimal choice of (G, E) implies that E / L is \(p\)-hypercyclically embedded in G / L. By Lemma 2.9, we have \(\Phi (E)=1\). By Lemma 2.10, we have that E contains a unique minimal G-invariant subgroup, say N.

(4)  Final contradiction.

Since \(\Phi (E)=1\), E is split over N (see [5, Chap. A, 9.10]). Hence \(E=[N]Y\) for some subgroup Y of E, and so \(E_{p}=[N]Y_{p}\) for some \(Y_{p}\in Syl_{p}(Y)\). Let U be a maximal subgroup of N such that U is \(E_{p}\)-invariant. Since \(d_{p}<|E_{p}|\), we may take a subgroup \(D=UV\) such that \(V\le Y_{p}\) and \(|D|=p^{d}\). Assume that \(D_{G}>1\). Then \(D\ge D_{G}\ge N\) because N is the unique minimal G-invariant subgroup of E, a contradiction. Hence \(D_{G}=1\). Now, the hypotheses imply that \(G=D[L]\) for some G-invariant subgroup L. Note that \(E\cap L=1\) contrary to \(|D|=|E|\). Thus \(E\cap L\) is a nontrivial normal subgroup of G, so \(U<N\le E\cap L\le L\), but then \(1<U\le D\cap L=1\), a contradiction. \(\square \)

Remark

The conclusion of Theorem A does not hold if we replace “c-normal” with “c-supplemented” in the hypothesis. One can take \(A_{5}\) for example. Obviously, every subgroup of \(A_{5}\) with order 5 is c-supplemented in \(A_{5}\), but \(A_5\) is not 5-hypercyclically embedded in itself.

Corollary 2.12

Let p be a fixed prime and \(G_p\) be a Sylow \(p\)-subgroup of a finite group G. Suppose that \(p^{d}\) is a prime power such that \(1<p^{d}\le \max (|G_{p}|/p,p)\). If all the subgroups of \(G_p\) with order \(p^{d}\) and \(2p^{d}\) (if a quaternion group is involved in \(G_{p}\)) are c-normal in G, then G is \(p\)-supersolvable.

Corollary 2.13

Let p be a fixed prime and E be a normal subgroup of a finite group G. Suppose that \(E_{p}\) is a Sylow \(p\)-subgroup of E and \(p^{d}\) is a prime power such that \(1<p^{d}\le \max (|E_{p}|/p,p)\). If G / E is \(p\)-supersolvable and all the subgroups of \(E_{p}\) with order \(p^{d}\) and \(2p^{d}\) (if a quaternion group is involved in \(E_{p}\)) are c-normal in G, then G is \(p\)-supersolvable.

Corollary 2.14

Let p be a fixed prime and E be a normal subgroup of a finite group G. Suppose that \(E_{p}\) is a Sylow \(p\)-subgroup of E and \(p^{d}\) is a prime power such that \(1<d\le \max (|E_{p}|/p,p)\). Suppose that \(N_{G}(E_{p})\) is \(p\)-nilpotent. If either \(E_{p}\) is abelian or every subgroup of \(E_{p}\) with order \(p^{d}\) and \(2p^{d}\) (if a quaternion group is involved in \(E_{p}\)) is c-normal in E, then G is \(p\)-nilpotent.

Proof

If \(E_{p}\) is abelian, then E is \(p\)-nilpotent by Burnside’s theorem. If \(E_{p}\) is not abelian, then E is \(p\)-supersolvable by Theorem A. In both cases, we have that \(E_{p}{{\mathrm{O}}}_{p'}(E)\) is a normal subgroup of G. By Frattini argument, \(G=N_{G}(E_{p}){{\mathrm{O}}}_{p'}(E)\). Note that \(N_{G}(E_{p})\) is \(p\)-nilpotent by hypotheses, and we have that G is \(p\)-nilpotent, as wanted. \(\square \)

Corollary 2.15

Let p be a fixed prime and \(G_p\) be a Sylow \(p\)-subgroup of a finite group G. Suppose that \(p^{d}\) is a prime power such that \(1<p^{d}\le \max (|G_{p}|/p,p)\). Suppose that \(N_{G}(G_{p})\) is \(p\)-nilpotent. If all the subgroups of \(G_p\) with order \(p^{d}\) and \(2p^{d}\) (if a quaternion group is involved in \(G_{p}\)) are c-normal in G, then G is \(p\)-nilpotent.

3 Some Applications

In this section, we give some applications to show that we can apply our results to get some known results.

Corollary 3.1

([2, Theorem 3.4]) Let \(\mathcal F\) be a saturated formation containing \(\mathcal U\). If all minimal subgroups and all cyclic subgroups with order 4 of \(G^{\mathcal F}\) are c-normal in G, then \(G\in \mathcal {F}\).

Proof

From Theorem A, we know that \(G^{\mathcal F}\) is \(p\)-hypercentrally embedded in G for all \(p\in \pi (G^{\mathcal {F}})\) and thus \(G^{\mathcal {F}} \le Z_{\mathcal {U}}(G)\). Since \(\mathcal F\) is a saturated formation containing \(\mathcal U\), we have that \(Z_{\mathcal {U}}(G)\le Z_{\mathcal {F}}(G)\). Consequently, \(G\in \mathcal {F}\) because \(G/G^{\mathcal {F}}\in \mathcal F\) and \(G^{\mathcal F}\le Z_{\mathcal {U}}(G)\le Z_{\mathcal {F}}(G)\). \(\square \)

The following lemma is evident.

Lemma 3.2

Let G be a group and p be a prime such that \((p-1,|G|)=1\). Then G is \(p\)-nilpotent if and only if G is \(p\)-supersolvable.

Corollary 3.3

([12, Theorem 0.1]) Let E be a normal subgroup of a group G of odd order such that G / E is supersolvable. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that \(1 < |D| < |P|\) and all subgroups H of P with order \(|H|=|D|\) are c-normal in G. Then G is supersolvable.

Proof

Let p be the minimal prime divisor of |E|. If \(E_{p}\) is cyclic, then E is \(p\)-nilpotent by [14, Lemma 2.8]. If \(E_{p}\) is not cyclic, then by Theorem A, E is \(p\)-supersolvable and thus \(p\)-nilpotent by Lemma 3.2. By repeating this argument, we know that E has a Sylow-tower and therefore E is solvable. Let p be any prime divisor of |E|. If \(E_{p}\) is cyclic, then E is \(p\)-hypercentrally embedded in G since now E is \(p\)-solvable. If \(E_{p}\) is not cyclic, then E is also \(p\)-hypercentrally embedded in G by Theorem A. Therefore we have \(E\le Z_{\mathcal {U}}(G)\). It follows that G is supersolvable since G / E is supersolvable and \(E\le Z_{\mathcal {U}}(G)\). \(\square \)

Corollary 3.4

([9, Theorem 3.1]) Let p be an odd prime dividing the order of a group G and P be a Sylow p-subgroup of G. If \(N_{G}(P)\) is \(p\)-nilpotent and every maximal subgroup of P is c-normal in G, then G is \(p\)-nilpotent.

Corollary 3.5

([9, Theorem 3.4]) Let p be the smallest prime dividing the order of a group G and P be a Sylow \(p\)-subgroup of G. If every maximal subgroup of P is c-normal in G, then G is \(p\)-nilpotent.

Proof

If \(|P|=p\), then G is \(p\)-nilpotent by [14, Lemma 2.8]. If \(|P|>p\), then by Corollary 2.12 G is \(p\)-supersolvable. Hence G is \(p\)-nilpotent by Lemma 3.2. \(\square \)

Corollary 3.6

([9, Theorem 3.6]) Let p be the smallest prime dividing the order of group G and P be a Sylow \(p\)-subgroup of G. If every minimal subgroup of \(P\cap G'\) is c-normal in G and when \(p=2\), either every cyclic subgroup of \(P\cap G'\) with order 4 is also c-normal in or P is quaternion free, then G is \(p\)-nilpotent.

Proof

By Theorem A, \(G'\) is \(p\)-hypercyclically embedded in G. Since \(G/G'\) is abelian, G is \(p\)-supersolvable. It then follows from Lemma 3.2 that G is \(p\)-nilpotent. \(\square \)

Corollary 3.7

([9, Corollary 3.9]) Let p be an odd prime dividing the order of a group G and P be a Sylow \(p\)-subgroup of G. If every minimal subgroup of \(P\cap G'\) is c-normal in G, then G is \(p\)-supersolvable.