1. Introduction

All groups considered in this paper will be finite. Two subgroups and of a group are said to permute if . It is easily seen that and permute if and only if is a subgroup of . We say, following Kegel [1], that a subgroup of is -quasinormal in if it permutes with every Sylow subgroup of . Chen [2] introduced the following concept: a subgroup of is said to be -semipermutable in if permutes with every Sylow -subgroup of with . Obviously, every -quasinormal subgroup of is an -semipermutable subgroup of . In contrast to the fact that every -quasinormal sub-group of is a subnormal subgroup of (see [1]), it does not hold in general that every -semipermutable subgroup of is a subnormal subgroup of . It suffices to consider the alternating group of degree 4.

Several authors have investigated the structure of a finite group when some information is known about some subgroups of prime power order in the group. Huppert [3] proved that a finite group is solvable provided that all subgroups of prime order are normal in . Buckley [4], proved that a group of odd order is supersolvable provided that all subgroups of prime order are normal in . Srinivasan [5], and proved that a finite group is supersolvable if the maximal subgroups of every Sylow subgroup of are normal in . Developing the result of Srinivasan, Ramadan [6] proved that if is a solvable group and the maximal subgroups of every Sylow subgroup of the Fitting subgroup of are normal in , then is supersolvable.

For a finite -group , we denote where .

Of late there has been a considerable interest to investigate the influence of the abelian subgroups of largest possible exponent of prime power order (we call such subgroups ALPE-subgroups) on the structure of the group. Asaad et al. [7] proved that if is a group such that for every prime and every Sylow -subgroup of , the ALPE-subgroups of (resp., ) are normal in , then is supersolvable. Ramadan [8] proved the following two results. (1) Let be a group such that for every prime and every Sylow -subgroup of , the ALPE-subgroups of (resp., ) are -quasinormal in , then is supersolvable. (2) Let be a normal subgroup of such that is supersolvable. If for every prime and every Sylow -subgroup of , the ALPE-subgroups of (resp., ) are -quasinormal in , then is supersolvable.

In this paper, we study the structure of a finite group under the assumption that certain subgroups of prime power order are -semipermutable in the group. We focus our attention on -semipermutability property of the ALPE-subgroups of a fixed ALPE-subgroup having maximal order of the Sylow subgroups of a finite group. Furthermore, we improve and extend the above-mentioned results by using the concept of -semipermutability.

2. Preliminaries

In this section, we give some results which will be useful in the sequal.

Lemma 2.1 (see [2, Lemmas 1 and 2]). Let be a group.(i)If is a -semipermutable subgroup of and is a subgroup of such that , then is -semipermutable in .(ii)Let be a set of primes, a normal -subgroup of , and a -subgroup of . If is -semipermutable in , then is -semipermutable in .

Lemma 2.2 (see [9, Lemma ]). Let be a -subgroup of ; for some prime . Then is -quasinormal in if and only if , where is the normal subgroup of generated by all -elements of .

Lemma 2.3. Let be a -subgroup of , is a prime. Then the following statements are equivalent:(i) is -quasinormal in ;(ii) and is -semipermutable in .

Proof. : Suppose that is -quasinormal in . So it follows by [1, Satz 1, page 209] that is subnormal in and then by [10, Lemma , page 28] that . Since is -quasinormal in , obviously, it is -semipermutable in . Thus (ii) holds.
: Since is -semipermutable in , then for every Sylow -subgroup of with . Clearly, is normal in and so . Thus . Applying Lemma 2.2, we have that is -quasinormal in . Thus (i) holdes.

Lemma 2.4 (see [7, Theorem 4, page 253]). Let be a normal -subgroup of . If the ALPE-subgroups of are normal in , then is supersolvably embedded in .

Lemma 2.5 (see [11, Lemma 3.8, page 2245]). Let be the smallest prime dividing the order of a group , and let be a Sylow -subgroup of . If , then is -nilpotent.

Lemma 2.6 (see [12, Lemma 2.6]). Let be a nontrivial normal subgroup of a group . If , then the Fitting subgroup of is the direct product of minimal normal subgroups of which are contained in .

Lemma 2.7 (see [13, Lemma , page 23]). Suppose that is a normal Sylow -subgroup of and that is supersolvable, where is a -Hall subgroup of . Then is supersolvable.

3. Main Results

Theorem 3.1. Let be the smallest prime dividing the order of a group , and let be a Sylow -subgroup of . Fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is -nilpotent.

Proof. Suppose that the theorem is false, and let be a counterexample of minimal order. We prove the following steps.
(1) If , Then Is -Nilpotent
It is clear to see by Lemma 2.1 that the ALPE-subgroups of are -semipermutale in , so that satisfies the hypothesis of the theorem. Thus, the minimality of yields that is -nilpotent.
(2) Is -Nilpotent
Suppose that is normal in . Let be an ALPE-subgroup of (in particular, we may take ). By hypothesis, is -semipermutable in and so by Lemma 2.3, we have that is -quasinormal in . Hence is a subgroup of , where is a Sylow -subgroup of with . Clearly, is a subnormal Hall subgroup of . Thus is normal in and hence is normal in as is abelian. Thus is supersolvably embedded in by Lemma 2.4 and so . Since by [14, page 34], it follows by Lemma 2.5 that is -nilpotent. Thus . Hence , so that , then is nilpotent by (1). Thus is -nilpotent and so is -nilpotent: a contradiction. Thus we may assume that . Then , in particular, . So, by the maximality of and we have by [15, Theorem 4.3, page 252] that is -nilpotent: a contradiction. Thus we may assume that . According to (1), we have that is -nilpotent.
(3)
If , we consider the quotient group . Clearly, is a Sylow -subgroup of and is an ALPE-Subgroup of having maximal order. By Lemma 2.1, the hypotheses are inherited over . Thus, the minimality of implies that is -nilpotent, hence is -nilpotent, which is a contradiction.
(4) , Where Is a Sylow -Subgroup of with
Since is not -nilpotent by [15, Theorem 4.5, page 253], there exists a subgroup of such that is not -nilpotent. But is -nilpotent by a similar argument of the proof of the step (2). Thus we may choose a subgroup of such that is not -nilpotent but is -nilpotent for every subgroup of with . It is easy to see that . If , it follows by (1) that is -nilpotent: a contradiction. Thus . This leads to and is -nilpotent for every subgroup of with . Now, by [15, Theorem 4.5, page 253] again, we see that is -nilpotent and therefore that is -solvable. Since is -solvable, for any with , there exists a Sylow -subgroup of such that by [15, Theorem 3.5, page 229]. If , then is -nilpotent by (1) and hence is -nilpotent. Thus . This leads to by [15, Theorem 3.2, page 228] as by (3), which is a contradiction. Thus .
(5) The Final Contradiction
Let be a minimal normal subgroup of such that . Clearly, and so by the maximality of . Hence . By hypothesis, for any Sylow -subgroup of with . It is easy to see that . Thus . If , then by (1), is -nilpotent. Hence is -nilpotent and so also does : a contradiction. Thus we may assume that . By the minimality of and since , we have that and so . If , then is -nilpotent by (1) and hence is -nilpotent. Thus and so . Thus by (4), is a -group and so by [14, Theorem 6.3, page 221], . Since , we have that which implies that is supersolvably embedded in and so clearly that . Thus, it is easy to see that the quotient group satisfies the hypothesis of the theorem by Lemma 2.1. Now, by the minimality of , we see that is -nilpotent. Since the class of all -nilpotent groups is a saturated formation, we have that is the unique minimal normal subgroup of and . Thus and hence by Lemma 2.6 and so by (3). Hence . Since is solvable, it follows by [15, Theorem 2.6, page 216] that : a contradiction. Thus we must have . Let be a Sylow -subgroup of . By (2), we have that . Hence . Thus , and, therefore, is -nilpotent by [15, Theorem 4.3, page 252]: a final contradiction.

We need the following result.

Theorem 3.2. Let be a saturated formation containing the class of supersolvable groups . Let be a normal Sylow -subgroup of a group such that . Fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then .

Proof. We treat the following two cases.
Case 1. .
Clearly, is a normal Sylow -subgroup of and is an ALPE-subgroup of having maximal order. By hypothesis and Lemma 2.1, the ALPE-subgroups of are -semipermutable in . Clearly,
Thus, our hypothesis carries over to and so by induction on the order of . Therefore, .
Case 2. .
Let be an ALPE-subgroup of . Then is -quasinormal in by Lemma 2.3 and hence by Lemma 2.2. Let . Then is normal in . Thus Lemma 2.4 implies that is supersolvably embedded in . Then, is supersolvable by [14, Lemma 7.15, page 35]. Clearly, , where is a Sylow -subgroup of . Let be a -subgroup of . Then is a group of automorphisms of . But , and consequently, acts trivially on . Then acts trivially on by [15, Theorem 3.4, page 179], that is, . It is easy to see that is subnormal in and so .  Hence .  Since is solvable, it follows by [15, Theorem 2.6, page 216] that : a contradiction. Hence must be a -group and so . Thus, is supersolvable which implies that is supersolvable by [16, Theorem 4]. Thus is supersolvable and therefore, is supersolvable by [17, Exercise , page 159]. Hence, .

As an immediate consequence of Theorem 3.2, we have the following theorem.

Corollary 3.3. Let be a normal Sylow -subgroup of a group such that is supersolvable. Fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is supersolvable.

We now prove the following theorem.

Theorem 3.4. Let be a group. For every prime and every Sylow -subgroup of , fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is supersolvable.

Proof. By repeated applications of Theorem 3.1, the group has a Sylow tower of supersolvable type. Hence has a normal Sylow -subgroup , where is the largest prime dividing the order of . By Lemma 2.1, our hypothesis carries over to . Thus is supersolvable by induction on the order of . Now, it follows from Corollary 3.3 that is supersolvable.

As an immediate consequence of Theorem 3.4, we have the following corollary.

Corollary 3.5 (Asaad et al. [7]). If is a group such that the ALPE-subgroups of every Sylow subgroup of are normal in , then is supersolvable.

Corollary 3.6 (Ramadan [8]). If is a group such that the ALPE-subgroups of every Sylow subgroup of are -quasinormal in , then is supersolvable.

We need the following Lemma.

Lemma 3.7. Let be a normal -subgroup of a group such that is supersolvable. Fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is supersolvable.

Proof. Let be a Sylow -subgroup of . We treat the following two cases.
Case 1. .
Then by Corollary 3.3, is supersolvable.
Case 2. .
Put , where . Since is supersolvable, it follows by [18, Theorem 5, page 5] that possesses supersolvable subgroups and such that and . Since and are supersolvable, it follows that and are supersolvable by induction on the order of . Since and , it follows again by [18, Theorem 5, page 5] that is supersolvable.

Now, we can prove the following theorem.

Theorem 3.8. Let be a normal subgroup of such that is supersolvable. For every prime dividing the order of and every Sylow -subgroup of , fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is supersolvable.

Proof. By Lemma 2.1, the ALPE-subgroups of are -semipermutable in . Hence is supersolvable by Theorem 3.4. Thus has a normal Sylow -subgroup , where is the largest prime dividing the order of . Since is characteristic in and , we have that . Clearly, is supersolvable. By Lemma 2.1, our hypothesis carries over to and hence is supersolvable by induction on the order of . Now, it follows from Lemma 3.7 that is supersolvable.

As an immediate consequence of Theorem 3.8, we have the following corollary.

Corollary 3.9 (Ramadan [8]). Let be a normal subgroup of a group such that is supersolvable. If the ALPE-subgroups of every Sylow subgroup of are -quasinormal in , then is supersolvable.

4. Similar Results

Following similar arguments to those of Theorem 3.1, it is possible to prove the following result.

Theorem 4.1. Let be the smallest prime dividing the order of a group and let be a Sylow -subgroup of . Fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is -nilpotent.

We prove the following lemma.

Lemma 4.2. Let be a normal -subgroup of a group such that is supersolvable. Fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is supersolvable.

Proof. Let be a Sylow -subgroup of . We treat the following two cases.
Case 1 . By [15, Theorem 2.1, page 221], there exists a -Hall subgroup , which is a complement to in . Hence is supersolvable. Since is characteristic in and , we have that . Clearly, is supersolvable. Thus, our hypothesis and Corollary 3.3 imply that is supersolvable. Therefore, is supersolvable by Lemma 2.7.
Case 2 . Put , where . Since is supersolvable, it follows by [18, Theorem 5, page 5] that possesses supersolvable subgroups and such that and . Since and are supersolvable, it follows that and are supersolvable by induction on the order of . Since and , it follows again by [18, Theorem 5, page 5] that is supersolvable.

By a similar proof to the proof of Theorem 3.4, we can prove the following theorem.

Theorem 4.3. Let be a group. For every prime and every Sylow -subgroup of an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is supersolvable.

As an immediate consequence of Theorem 4.3, we have the following corollary.

Corollary 4.4 (Asaad et al. [7]). If is a group such that for every prime and every Sylow -subgroup , the ALPE-subgroups of are normal in , then is supersolvable.

Corollary 4.5 (Ramadan [8]). If is a group such that for every prime and every Sylow -subgroup , the ALPE-subgroups of are -quasinormal in , then is supersolvable.

We can now prove the following corollary.

Corollary 4.6. Let be a normal subgroup of such that is supersolvable. For every prime dividing the order of and every Sylow -subgroup of , fix an ALPE-subgroup of having maximal order. If the ALPE-subgroups of are -semipermutable in , then is supersolvable.

Proof. By Lemma 2.1, the ALPE-subgroups of are -semipermutable in . Hence is supersolvable by Theorem 4.3. Thus has a normal Sylow -subgroup , where is the largest prime dividing the order of . Since is characteristic in and , we have that . Clearly, is supersolvable. By Lemma 2.1, the hypothesis of our theorem carries over to . Thus is supersolvable by induction on the order of and it follows that is supersolvable by Lemma 4.2.

Remarks 4.7. (a) The converse of Theorem 3.4 is not true. For example, set , where and . Clearly, is supersolvable and has an abelian Sylow 3-subgroup of exponent 3. It is easy to check that contains a subgroup of order 3 which fails to be -semipermutable in .
(b) Theorem 4.3 is not true when the smallest prime dividing the order of is even and , where is a Sylow -subgroup of . For example, if is the quaternion group , is a cyclic group of order 9 with generator , and the action of on is given by , then the semidirect product of by is a group of even order in which every subgroup of prime order is -semipermutable. Clearly, the semidirect product of by is not supersolvable (see Buckley [4, Examples ]).