Local Classes and Pairwise Mutually

Permutable Products of Finite Groups

A. Ballester-Bolinches, J. C. Beidleman, H. Heineken and M. C. Pedraza-Aguilera

Received: January 28, 2009 Revised: November 19, 2009

Communicated by Ulf Rehmann

Abstract. The main aim of the paper is to present some results about products of pairwise mutually permutable subgroups and local classes.

Keywords and Phrases: mutually permutable, local classes, p-soluble groups, p-supersolubility, finite groups

1 Introduction

If A and B are subgroups of a group G, the product AB of A and B is defined to be the subset of all elements of G with the form ab, where a ∈ A, b ∈ B. It is well known that AB is a subgroup of G if and only if AB = BA, that is, if the subgroups A and B permute. Should it happen that AB coincides with the group G, with the result that G = AB = BA, then G is said to be factorized by its subgroups A and B. More generally, a group G is said to be the product of its pairwise permutable subgroups G_{1}, G_{2}, …, G_{n} if G = G_{1}G_{2} … G_{n} and G_{i}G_{j} = G_{j}G_{i} for all integers i and j with i, j ∈ {1, 2, …, n}. This implies that for every choice of indices 1 ≤ i_{1} ≤ i_{2} ≤ … ≤ i_{k} ≤ n, the product G_{i_{1}}G_{i_{2}} … G_{i_{k}} is a subgroup of G. Groups which are product of two of its subgroups have played a significant part in the theory of groups over the past sixty years. Among the central problems considered the following ones are of interest to us:

Let the group G = G_{1}G_{2} … G_{n} be the product of its pairwise permutable subgroups G_{1}, G_{2}, …, G_{n} and suppose that the factors G_{i}, 1 ≤ i ≤ n, belong to a class of groups X. When does the group G belong to X?. How does the structure of the factors G_{i}, 1 ≤ i ≤ n affect the structure of the group G?.

Obviously, if G_{i}, 1 ≤ i ≤ n, are finite, then the group G is finite. However not many properties carry over from the factors of a factorized group to the group itself. Indeed if one thinks about properties such as solubility, supersolubility, or nilpotency, one soon realizes the difficulty of using factorization to obtain information about the structure of the whole group. Two well known examples support the above claim: there exist non abelian groups which are products of two abelian subgroups and every finite soluble group is the product of pairwise permutable nilpotent subgroups. However a prominent result by Itô shows that every product of two abelian groups is metabelian, and an important result of Kegel and Wielandt shows the solubility of every finite group G = G_{1}G_{2} … G_{n} which is the product of pairwise permutable nilpotent subgroups G_{i}, 1 ≤ i ≤ n. In the much more special case when G_{i}, 1 ≤ i ≤ n, are normal nilpotent subgroups of G, the product G_{1}G_{2} … G_{n} is nilpotent. This is a well known result of Fitting. However, if G_{1}, G_{2}, …, G_{n} are normal supersoluble subgroups of G, the product G_{1}G_{2} … G_{n} is not supersoluble in general even in the finite case (see [1]). Consequently it seems reasonable to look into these problems under additional assumptions. In this context, assumptions on permutability connections between the factors turn out to be very useful. One of the most important ones is the mutual permutability introduced by Asaad and Shaalan in [1]. We say that two subgroups A and B of a group G are mutually permutable if A permutes with every subgroup of B and B permutes with every subgroup of A. If G = AB and A and B are mutually permutable, then G is called a mutually permutable product of A and B. More generally, a group G = G_{1}G_{2} … G_{n} is said to be the product of the pairwise mutually permutable subgroups G_{1}, G_{2}, …, G_{n} if G_{i} and G_{j} are mutually permutable subgroups of G for all i, j ∈ {1, 2, …, n}. Asaad and Shaalan ([1]) proved that if G is a mutually permutable product of the subgroups A and B and A and B are finite and supersoluble, then G is supersoluble provided that either G′, the derived subgroup of G, is nilpotent or A or B is nilpotent. This result was the beginning of an intensive study of such factorized groups (see, for instance, [2, 4, 6, 9] and the papers cited therein). The extension of the above results on mutually permutable products of two subgroups to general pairwise mutually permutable products turns out to be difficult in many cases. Carocca proved (see [10]) that if the derived subgroup of a pairwise mutually permutable product of supersoluble subgroups is nilpotent, then the group G is supersoluble. However a pairwise mutually permutable product of supersoluble groups in which one of them is nilpotent is not supersoluble in general (see [4, Example]). Nevertheless in [4] we obtained that if G is the pairwise mutually permutable product of supersoluble subgroups with all factors but one nilpotent, then the group is supersoluble. Some interesting results on pairwise mutually permutable products arise when the factors belong to some classes of finite groups which are defined in terms of permutability. They are the class of P ST -groups, or finite groups G in which every subnormal subgroup of G permutes with every Sylow subgroup of G, the class of P T -groups, or finite groups in which every subnormal subgroup is a permutable subgroup of the group, the class of T -groups, or groups in which every subnormal subgroup is normal, and the class of Ygroups, or finite groups G for which for every subgroup H and for all primes q dividing the index |G : H| there exists a subgroup K of G such that H is contained in K and |K : H| = q, and their corresponding local versions (see [2, 3]).

The main purpose of this article is to take this program of research a step further by analyzing the structure of the pairwise mutually permutable products whose factors belong to some local classes of finite groups closely related to the classes of all T -groups and Y-groups.

Therefore in the sequel all groups considered are finite.

2 The class C̄_{p} and pairwise mutually permutable products

Throughout this section, p will be a prime. Recall that a group G satisfies property C_{p}, or G is a C_{p}-group, if each subgroup of a Sylow p-subgroup P of G is normal in the normalizer N_{G}(P). This class of groups was introduced by Robinson in his seminal paper [14] as a local version of the class of all soluble T -groups. In fact, he proved there that a group G is a soluble T -group if and only if G is a C_{p}-group for all primes p. In [7] the second and third authors introduce and analyze an interesting class of groups closely related to the class of all T -groups. A group G is a T_{1}-group if G/Z_{∞}(G) is a T -group. Here Z_{∞}(G) denotes the hypercenter of G, that is, the largest normal subgroup of G having a G-invariant series with central G-chief factors. The local version of the class T_{1} in the soluble universe is the class C̄_{p} introduced and studied in [8]:

Definition 1. Let G be a group and let Z_{p}(G) be the Sylow p-subgroup of Z_{∞}(G). A group satisfies C̄_{p} if and only if G/Z_{p}(G) is a C_{p}-group.

Theorem A ([8]) A group G is a soluble T_{1}-group if and only if G is a C̄_{p}-group for all primes p.

The objective of this section is to analyze the behaviour of pairwise mutually permutable products with respect to the class C̄_{p}.

We begin with some results concerning the classes C_{p} and C̄_{p}.

Lemma 1. [8, Lemma 2] Let p be a prime. Then:

(i) C_{p} is a subgroup-closed class.

(ii) Let M be a normal p′-subgroup of a group G. If G/M is a C_{p}-group, then so is G.

(iii) If G is a C_{p}-group and N is a normal subgroup of G, then G/N is a C_{p}-group.

Lemma 2. Let G be a C̄_{p}-group and let N be a normal subgroup of G. Then G/N is a C̄_{p}-group.

Proof Let Z_{p}(G) be the Sylow p-subgroup of Z_{∞}(G). Since G/Z_{p}(G) is a C_{p}-group, it follows that G/Z_{p}(G)N is a C_{p}-group by Lemma 1. Let H/N denote the Sylow p-subgroup of Z_{∞}(G/N). Since Z_{p}(G)N/N is contained in H/N, we have that (G/N)/(H/N) is isomorphic to a quotient of G/Z_{p}(G)N. By Lemma 1, (G/N)/(H/N) is a C_{p}-group. Therefore G/N is a C̄_{p}-group.

Recall that a group G is said to be p-supersoluble if it is p-soluble and every p-chief factor of G is cyclic. It is rather clear that the derived subgroup of a p-supersoluble group is p-nilpotent and, if p = 2, the group itself is 2-nilpotent.

Lemma 3. [8, Lemma 3] Let G be a p-soluble group. If G is a C̄_{p}-group, then G is p-supersoluble.

The main aim of this section is to show that pairwise mutually permutable products of p-soluble C̄_{p}-groups are p-supersoluble.

Theorem 1. Let G = G_{1}G_{2} … G_{k} be the pairwise mutually permutable product of the subgroups G_{1}, G_{2}, …, G_{k}. If G_{i} is a p-soluble C̄_{p}-group for every i ∈ {1, 2, …, k}, then G is p-supersoluble.

Proof Assume that the theorem is false, and let G be a counterexample with minimal order. By [4, Theorem 1], G is p-soluble. If p = 2, then G_{i} is 2nilpotent for all i = 1, 2, …, k and so G is 2-supersoluble by [4, Theorem 3]. This contradiction implies that p is odd. Note, that the hypotheses of the theorem are inherited by all proper quotients of G. Therefore the minimal choice of G yields G/N p-supersoluble for every minimal normal subgroup N of G. Since the class of p-supersoluble groups is a saturated formation, it follows that G has a unique minimal normal subgroup, say N, G/N is p-supersoluble, the Frattini subgroup of G is trivial and then N = C_{G}(N) = F (G) = O_{p}(G). Moreover, N is an elementary abelian p-group of rank greater than 1. By Lemma 3, G_{i} is p-supersoluble, for all i = 1, 2, …, k. Consequently (G_{i})′ is p-nilpotent. Furthermore, by [4, Lemma 1(iii)], we have that (G_{i})′ is a subnormal subgroup of G for all i. Since O_{p′}(G) = 1, it follows that (G_{i})′ is a p-group and then G_{i} is supersoluble for all i. Then G_{i} is a Sylow tower group with respect to the reverse natural ordering of the prime numbers for all i. Applying [4, Corollary 1], G is a Sylow tower group with respect to the reverse natural ordering of the prime numbers. Therefore p is the largest prime dividing the order of G and F (G) = N is the Sylow p-subgroup of G. Now we observe the following facts:

(i) For each i ∈ {1, 2, …, k}, either N ≤ G_{i} or N ∩ G_{i} = 1. Put R := N ∩G_{i}, and assume that R ≠ 1. Let H_{j} be a Hall p′-subgroup of G_{j} (such Hall subgroups exist since G_{j} is soluble). Then R = G_{i}H_{j} ∩ N, so G_{j} ≤ NH_{j} ≤ N_{G}(R) for every j. Hence R is a normal subgroup of G and R = N.

(ii) Let N ≤ G_{i}, with i ∈ {1, 2, …, k}. Then Z_{∞}(G_{i}) = Z(G_{i}), N = Z(G_{i})× [N, G_{i}], and every subgroup of [N, G_{i}] is G_{i}-invariant. Clearly N ≤ F (G_{i}) and O_{p′}(F (G_{i})) ≤ C_{G}(N) = N. Thus F (G_{i}) = N. Therefore Z_{∞}(G_{i}) ≤ N, and Z_{∞}(G_{i}) = Z(G_{i}), since G_{i}/C_{G_{i}}(Z_{∞}(G_{i})) is a p-group (G_{i} stabilizes a series of subgroups of Z_{∞}(G_{i}), see [11, A, 12.4]) and N is a Sylow p-subgroup of G_{i}. Moreover, N = Z(G_{i})×[N, G_{i}] since G_{i}/N is a p′-group. As G_{i} ∈ C̄_{p} and Z(G_{i}) = Z_{∞}(G_{i}), G_{i} normalizes every subgroup of [N, G_{i}].

(iii) Let N ≤ G_{i} with i ∈ {1, 2, …, k}, then every y ∈ G_{i} \ N induces a non-trivial GF (p)-scalar multiplication on [N, G_{i}]; in particular C_{N}(y) = Z(G_{i}) and G_{i}/N is cyclic. Note that G/N acts faithfully on N. So y induces a non-trivial linear mapping on the GF (p)-space [N, G_{i}] that leaves invariant every subspace. It is well-known that these mappings come from multiplication with an element of GF (p).

(iv) Let N ≤ G_{i} and N ≤ G_{j} with i ∈ {1, 2, …, k}. Suppose that N_{G_{i}}(Z(G_{j})) ≰ N. Then G_{i} ≤ N_{G}(Z(G_{j})). Put R := N_{G_{i}}(Z(G_{j})). By (iii), Z(G_{j}) = (Z(G_{j}) ∩ Z(G_{i})) × [Z(G_{j}), R], and [Z(G_{j}), R] ≤ [N, G_{i}]. Thus by (ii), Z(G_{j}) is G_{i}-invariant.

(v) Suppose that N ≤ G_{i} and G_{j} ≤ N_{G}(Z(G_{i})). Then [G_{i}, G_{j}] ≤ N; in particular, if N ≤ G_{j}, G_{i} ≤ N_{G}(Z(G_{j})). Put H := G_{i}G_{j}. Then H/N is a p′-group. By Maschke’s Theorem there exists an H-invariant complement N_{0} for Z(G_{i}) in N. By (iii) N_{0} = [N, G_{i}] and [G_{i}, G_{j}] ≤ C_{H}(N_{0}). Since also [G_{i}, G_{j}] ≤ C_{H}(Z(G_{i})), it follows that [G_{i}, G_{j}] ≤ C_{H}(N) ≤ N. Moreover if N ≤ G_{j} we have that G_{j}^{H} = G_{j}^{G_{i}} = G_{j}[G_{i}, G_{j}] = G_{j}, that is, G_{j} is a normal subgroup of H and then G_{i} normalizes Z(G_{j}).

(vi) Let R ≤ N ∩ G_{i} and G_{j} ∩ N = 1. Then G_{j} normalizes R.

Since RG_{j} is a subgroup of G, RG_{j} ∩ N = R is a normal subgroup of RG_{j}.

(vii) Suppose that N ≤ G_{j}. Then Z(G_{j}) is a normal subgroup of G. We may assume that there exists i ∈ {1, 2, …, k} such that G_{i} ≰ N_{G}(Z(G_{j})). In particular Z(G_{j}) ≠ 1, and Z(G_{j}) ≤ N by (ii). Now the application of (vi) yields G_{i}∩N ≠ 1 and so by (i) also N ≤ G_{i}. Moreover, G_{i} ∉ C_{p} and so Z_{∞}(G_{i}) ≠ 1. Applying (ii) Z_{∞}(G_{i}) = Z(G_{i}) ≠ 1, and by (v) G_{j} ≰ N_{G}(Z(G_{i})). Hence the situation is completely symmetric in i and j. Put H := G_{i}G_{j}. We first show:

(*) G_{i} ∩ G_{j} = N, and N_{H}(RN) = G_{k} for every k ∈ {i, j} and R ≤ G_{k} with R ≰ N. Since G_{k}/N is cyclic by (iii), RN is a normal subgroup of G_{k} and so N_{H}(RN) = N_{G_{t}}(RN)G_{k}, where {t, k} = {i, j}. Now (iv) yields N_{G_{t}}(RN) ≤ N. This shows that G_{i} ∩ G_{j} = N and N_{H}(RN) = G_{k}. As a consequence of (*), G_{i}/N and G_{j}/N have trivial intersection, therefore H/N = (G_{i}/N)(G_{j}/N) is the totally permutable product of G_{i}/N and G_{j}/N (see [6, Lemma 1]), that is, every subgroup of G_{i}/N permutes with every subgroup of G_{j}/N. Thus there exists RN/N a minimal normal subgroup of H/N contained in G_{i}/N or in G_{j}/N (see [10]), suppose RN/N ≤ G_{i}/N without loss of generality. Then N_{H}(RN) = H. On the other hand, by (*) N_{H}(RN) = G_{i}. But then G_{j} ≤ G_{i}, a contradiction since G_{j} ≰ N_{G}(Z(G_{i})).

Since not all the factors G_{i} are p′-groups, there exists G_{i} with N ≤ G_{i}. It suffices to show that every subgroup R of N is normal in G. By (i) and (vi) every G_{j} with N ≰ G_{j} normalizes R. On the other hand, by (vii) for every G_{j} with N ≤ G_{j} either N = Z(G_{j}) = G_{j} or Z(G_{j}) = 1. In the first case obviously G_{j} ≤ N_{G}(R). In the second case G_{j} ∈ C_{p} and again G_{j} ≤ N_{G}(R). Consequently | N |= p, the final contradiction.

Combining Theorems A and 1 we have:

Corollary 1. Let G = G_{1}G_{2} … G_{k} be a product of the pairwise mutually permutable soluble T_{1}-groups G_{1}, G_{2}, …, G_{k}. Then G is supersoluble.

3 The class Ẑ_{p} and pairwise mutually permutable products

Another interesting class of groups closely related to T -groups is the class T_{0} of all groups G whose Frattini quotient G/Φ(G) is a T -group. This class was introduced in [15] and studied in [12, 13, 15]. The procedure of defining local versions in order to simplify the study of global properties has also been successfully applied to the study of the classes T_{0} ([12]) and Y ([3]).

Definition 2. Let p be a prime and let G be a group.

(i) ([12]) Let Φ(G)_{p} be the Sylow p-subgroup of the Frattini subgroup of G. G is said to be a Ĉ_{p}-group if G/Φ(G)_{p} is a C_{p}-group.

(ii) ([3, Definition 11]) We say that G satisfies Z_{p} or G is a Z_{p}-group when for every p-subgroup X of G and for every power of a prime q, q^{m}, dividing | G : XO_{p′}(G) |, there exists a subgroup K of G containing XO_{p′}(G) such that | K : XO_{p′}(G) |= q^{m}.

It is rather clear that the class of all Ĉ_{p}-groups is closed under taking epimorphic images and all p-soluble groups belonging to Ĉ_{p} are p-supersoluble. Moreover:

Theorem B ([12]) A group G is a soluble T_{0}-group if and only if G is a Ĉ_{p}-group for all primes p.

In the following two results we gather some useful properties of the Z_{p}-groups.

Lemma 4. Let G be a group.

(i) If G is a p-soluble Z_{p}-group, then G is p-supersoluble. [3, Lemma 20]

(ii) If G is a p-soluble Z_{p}-group and N is a normal subgroup of G, then G/N is a Z_{p}-group. [3, Lemma 18]

(iiii) Let G be a soluble group. G is a Y-group if and only if G is a Z_{p}-group for every prime p. [3, Theorem 15]

Theorem C [3, Theorem 13] Let p be a prime and G a p-soluble group. Then G satisfies Z_{p} if and only if G satisfies one of the following conditions:

(1) G is p-nilpotent.

(2) G(p)/O_{p′}(G(p)) is a Sylow p-subgroup of G/O_{p′}(G(p)) and for every psubgroup H of G(p), we have that G = N_{G}(H)G(p).

Here G(p) denotes the p-nilpotent residual of G, that is, the smallest normal subgroup of G with p-nilpotent quotient.

The results of [5] show that the class C_{p} is a proper subclass of the class Z_{p}.

In [2, Theorem 16] it is proved that a pairwise mutually permutable product of Y-groups is supersoluble. There it is asked whether a pairwise mutually permutable product of Z_{p}-groups is p-supersoluble. In this section, we answer to this question affirmatively. In fact, the main purpose here is to study pairwise mutually permutable products whose factors belong to some class of groups closely related to Z_{p}-groups.

Definition 3. Let p be a prime, let G be a group and let Φ(G)_{p} be the Sylow p-subgroup of the Frattini subgroup of G. G is said to be a Ẑ_{p}-group if G/Φ(G)_{p} is a Z_{p}-group.

Lemma 5. Let p be a prime and M a normal subgroup of G. If G is a Ẑ_{p}-group, then G/M is a Ẑ_{p}-group.

Proof Assume that G is a Ẑ_{p}-group. Then G/Φ(G)_{p} is a Z_{p}-group. Since Φ(G)_{p}M/M is contained in Φ(G/M)_{p} = L/M and the class of all Z_{p}-groups is closed under taking epimorphic images, we have that G/L belongs to Z_{p}. This is to say that G/M is a Ẑ_{p}-group.

Since the class of all p-supersoluble groups is a saturated formation and, by Lemma 4(i), every p-soluble Z_{p}-group is p-supersoluble, we have:

Lemma 6. Let p be a prime and let G be a p-soluble group. If G is a Ẑ_{p}-group, then G is p-supersoluble.

The main result of this section shows that pairwise mutually permutable products of Ẑ_{p}-groups are p-supersoluble.

Theorem 2. Let G = AB be the mutually permutable product of the psupersoluble group A and the p-soluble Ẑ_{p}-group B. Then G is p-supersoluble.

Proof Assume that the result is false, and let G be a counterexample of minimal order. Applying [4, Theorem 1], G is p-soluble. Let N be a minimal normal subgroup of G. Then G/N is the mutually permutable product of the subgroups AN/N and BN/N. Moreover, AN/N is p-supersoluble and BN/N is a p-soluble Ẑ_{p}-group by Lemma 5. The minimality of G implies that G/N is p-supersoluble. Since p-supersoluble groups is a saturated formation, it follows that G has a unique minimal normal subgroup, N say. Moreover N is an elementary abelian p-group of rank greater than 1 and N = C_{G}(N) = F (G) = O_{p}(G). Note further that, by Lemma 6, A and B are p-supersoluble. Applying [6, Lemma 1(vii)], we have that A and B either cover or avoid N. If A and B both avoid N, then |N| = p by [6, Lemma 2] and G is p-supersoluble. This contradiction allows us to assume that N ≤ A. Suppose that B ∩ N = 1 and let X be a minimal normal subgroup of A such that X ≤ N. Then |X| = p and XB ∩ N = X is a normal subgroup of XB. It means that B normalizes X and so X is a normal subgroup of G. This would imply that G is p-supersoluble, contrary to our supposition. We obtain also a contradiction if we assume N ≤ B and A ∩ N = 1. Therefore we may suppose that N ≤ A ∩ B. Note that, by [4, Theorem 3], neither A nor B is p-nilpotent. On the other hand, by [6, Theorem 1], we have that A′ and B′ are subnormal subgroups of G. Since they are p-nilpotent and O_{p′p}(G) = N, it follows that ⟨A′, B′⟩ ≤ N. Let 1 ≠ B(p) be the p-nilpotent residual of B. Then B(p) ≤ B′ ≤ N. Now observe that O_{p′}(B) = 1 and B is p-closed. Then it is an elementary fact that Φ(B) = Φ(O_{p}(B)) = Φ(B)_{p}. Since B is not p-nilpotent, Theorem C gives B(p) ∈ Syl_{p}(B), so N = B(p) and Φ(B) = Φ(N) = 1. In particular B ∈ Z_{p} and by Theorem C every subgroup of N is normal in B. Therefore, if X is a minimal normal subgroup of A contained in N, we have that X is a normal subgroup of G of order p. Consequently, G is p-supersoluble, the final contradiction.

Theorem 3. Let G = G_{1}G_{2} … G_{n} be the pairwise mutually permutable product of the subgroups G_{1}, G_{2}, …, G_{n}. If G_{i} is a p-soluble Ẑ_{p}-group for every i, then G is p-supersoluble.

Proof Assume that the theorem is false, and among the counterexamples with minimal order choose one G = G_{1}G_{2} … G_{n} such that the sum |G_{1}| + |G_{2}| + … + |G_{n}| is minimal. By Theorem 2, we have n > 2. Moreover, by [4, Theorem 1], G is p-soluble. It is rather clear that the hypotheses of the theorem are inherited by all proper quotients of G. Hence G contains a unique minimal normal subgroup, N say, N is not cyclic, G/N is p-supersoluble and N = C_{G}(N) = O_{p}(G). Hence O_{p′}(G) = 1. By Lemma 6, G_{i} is p-supersoluble and hence G′_{i} is p-nilpotent for every i. Applying Lemma 1(iii) of [4], we have that G′_{i} is a subnormal subgroup of G for each i ∈ {1, 2, …, n}. Hence G′_{i} is contained in N and so G_{i} is supersoluble for each i ∈ {1, 2, …, n}. By [4, Corollary 1], G is a Sylow tower group with respect to the reverse natural ordering of the prime numbers, p is the largest prime divisor of |G| and N is the Sylow p-subgroup of G. Let i ∈ {1, 2, …, n} such that p divides |G_{i}|. Then N ∩ G_{i} is the non-trivial Sylow p-subgroup of G_{i}. Let j ∈ {1, 2, …, n} such that j ≠ i. Then G_{i}(G_{j})_{p′} is a subgroup of G and N ∩G_{i} is a Sylow p-subgroup of G_{i}(G_{j})_{p′}. Since G_{i}(G_{j})_{p′} is a Sylow tower group with respect to the reverse natural ordering of the prime numbers, it follows that N ∩G_{i} is normal in G_{i}(G_{j})_{p′}. This implies that N ∩G_{i} is a normal subgroup of G and so N = N ∩ G_{i} is contained in G_{i}. Assume that there exists j ∈ {1, 2, …, n} such that p does not divide |G_{j}|. We may assume without loss of generality j = 1. Then G′_{1} = 1, that is, G_{1} is an abelian p′-group, and T = G_{2}G_{3} … G_{n} is p-supersoluble by the choice of G. Let R be a minimal normal subgroup of T contained in N. Then |R| = p. Moreover, G_{1}R is a subgroup of G because N is contained in some of the factors G_{l}, l > 1. Hence G_{1}R ∩ N = R is a normal subgroup of G_{1}R. Hence R is a normal subgroup of G and so N = R. This is a contradiction. Therefore p divides the order of G_{i} for every i ∈ {1, 2, …, n}. Consequently, N is contained in G_{i} for every i ∈ {1, 2, …, n}. Consider now W = G_{2}G_{3} … G_{n}. Then W is p-supersoluble. Let X be a minimal normal subgroup of W contained in N. Then |X| = p. Recall that G_{1} is a Ẑ_{p}-group. Assume that G_{1}/Φ(G_{1})_{p} is p-nilpotent. Then G_{1} is p-nilpotent. Since N is self-centralizing in G, it follows that G_{1} = N. Suppose that G_{1}/Φ(G_{1})_{p} satisfies condition (2) of Theorem C. Then we can argue as in the proof of Theorem 2 to obtain that N = G_{1}(p), the p-nilpotent residual of G_{1}, and Φ(G_{1})_{p} = 1. Consequently every subgroup of N is normal in G_{1}. In both cases, we have that G_{1} normalizes X. It means that N = X, the final contradiction.

Applying Theorems C and 3 we have:

Corollary 2. Let G = G_{1}G_{2} … G_{n} be a group such that G_{1}, G_{2}, …, G_{n} are pairwise mutually permutable subgroups of G. If all G_{i} are p-nilpotent, then G is p-supersoluble.

Since every Z_{p}-group is a Ẑ_{p}-group, we can apply Lemma 4(iii) and Theorem 3 to obtain the following:

Corollary 3. [2, Theorem 16] Let G = G_{1}G_{2} … G_{n} be a group such that G_{1}, G_{2}, …, G_{n} are pairwise mutually permutable subgroups of G. If all G_{i} are Y-groups, then G is supersoluble.

Finally, applying Theorems B and 3, we have:

Corollary 4. Let G = G_{1}G_{2} … G_{n} be a group such that G_{1}, G_{2}, …, G_{n} are pairwise mutually permutable subgroups of G. If all G_{i} are soluble T_{0}-groups, then G is supersoluble.

Acknowledgements

The authors wish to thank the referee for his/her clever reading of the paper and his/her valuable suggestions which helped to improve our results.

The first and fourth authors are supported by Proyecto MTM2007-68010-C03-02 (MEC) and FEDER (European Union).

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A. Ballester-Bolinches Departament d’Àlgebra Universitat de València Dr. Moliner 50, 46100 Burjassot València (Spain) Adolfo.Ballester@uv.es J. C. Beidleman Department of Mathematics University of Kentucky Lexington, Kentucky 40506-0027 U.S.A. clark@ms.uky.edu

H. Heineken Institut für Mathematik Universität Würzburg Am Hubland 97074 Würzburg (Germany) heineken@mathematik.uniwuerzburg.de M. C. Pedraza-Aguilera Instituto Universitario de Matemática Pura y Aplicada Universidad Politécnica de Valencia Camino de Vera, 46022 Valencia(Spain) mpedraza@mat.upv.es 266

Documenta Mathematica 15 (2010)