Finite Groups Whose Certain Subgroups of Prime Power Order Are 𝑆-Semipermutable

Abstract

Let 𝐺 be a finite group. A subgroup 𝐻 of 𝐺 is said to be S-semipermutable in 𝐺 if 𝐻 permutes with every Sylow 𝑝-subgroup of 𝐺 with (𝑝,|𝐻|)=1. In this paper, we study the influence of S-permutability property of certain abelian subgroups of prime power order of a finite group on its structure.

1. Introduction

All groups considered in this paper will be finite. Two subgroups 𝐻 and 𝐾 of a group 𝐺 are said to permute if 𝐻𝐾=𝐾𝐻. It is easily seen that 𝐻 and 𝐾 permute if and only if 𝐻𝐾 is a subgroup of 𝐺. We say, following Kegel [1], that a subgroup of 𝐺 is 𝑆-quasinormal in 𝐺 if it permutes with every Sylow subgroup of 𝐺. Chen [2] introduced the following concept: a subgroup 𝐻 of 𝐺 is said to be 𝑆-semipermutable in 𝐺 if 𝐻 permutes with every Sylow 𝑝-subgroup of 𝐺 with (𝑝,|𝐻|)=1. Obviously, every 𝑆-quasinormal subgroup of 𝐺 is an 𝑆-semipermutable subgroup of 𝐺. In contrast to the fact that every 𝑆-quasinormal sub-group of 𝐺 is a subnormal subgroup of 𝐺 (see [1]), it does not hold in general that every 𝑆-semipermutable subgroup of 𝐺 is a subnormal subgroup of 𝐺. 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 𝐺 is solvable provided that all subgroups of prime order are normal in 𝐺. Buckley [4], proved that a group 𝐺 of odd order is supersolvable provided that all subgroups of prime order are normal in 𝐺. Srinivasan [5], and proved that a finite group 𝐺 is supersolvable if the maximal subgroups of every Sylow subgroup of 𝐺 are normal in 𝐺. Developing the result of Srinivasan, Ramadan [6] proved that if 𝐺 is a solvable group and the maximal subgroups of every Sylow subgroup of the Fitting subgroup 𝐹(𝐺) of 𝐺 are normal in 𝐺, then 𝐺 is supersolvable.

For a finite 𝑝-group 𝑃, we denote Ω(𝑃)=Ω₁(𝑃)if𝑝>2,Ω(𝑃)=⟨Ω₁(𝑃),Ω₂(𝑃)⟩if𝑝=2,(1.1) where Ω_𝑖(𝑃)=⟨𝑥∈𝑃∶|𝑥|=𝑝^𝑖⟩.

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 𝐺 is a group such that for every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺_𝑝 of 𝐺, the ALPE-subgroups of 𝐺_𝑝 (resp., Ω(𝐺_𝑝)) are normal in 𝐺, then 𝐺 is supersolvable. Ramadan [8] proved the following two results. (1) Let 𝐺 be a group such that for every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺_𝑝 of 𝐺, the ALPE-subgroups of 𝐺_𝑝 (resp., Ω(𝐺_𝑝)) are 𝑆-quasinormal in 𝐺, then 𝐺 is supersolvable. (2) Let 𝐾 be a normal subgroup of 𝐺 such that 𝐺/𝐾 is supersolvable. If for every prime 𝑝 and every Sylow 𝑝-subgroup 𝐾_𝑝 of 𝐾, the ALPE-subgroups of 𝐾_𝑝 (resp., Ω(𝐾_𝑝)) are 𝑆-quasinormal in 𝐺, then 𝐺 is supersolvable.

In this paper, we study the structure of a finite group under the assumption that certain subgroups of prime power order are 𝑆-semipermutable in the group. We focus our attention on 𝑆-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 𝑆-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 𝐺 be a group.(i)If 𝐻 is a 𝑆-semipermutable subgroup of 𝐺 and 𝐾 is a subgroup of 𝐺 such that 𝐻≤𝐾≤𝐺, then 𝐻 is 𝑆-semipermutable in 𝐾.(ii)Let 𝜋 be a set of primes, 𝑁 a normal 𝜋^-subgroup of 𝐺, and 𝐻 a 𝜋-subgroup of 𝐺. If 𝐻 is 𝑆-semipermutable in 𝐺, then 𝐻𝑁/𝑁 is 𝑆-semipermutable in 𝐺/𝑁.

Lemma 2.2 (see [9, Lemma A]). Let 𝐻 be a 𝑝-subgroup of 𝐺; for some prime 𝑝. Then 𝐻 is 𝑆-quasinormal in 𝐺 if and only if 𝑂^𝑝(𝐺)≤𝑁_𝐺(𝐻), where 𝑂^𝑝(𝐺) is the normal subgroup of 𝐺 generated by all 𝑝^-elements of 𝐺.

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

Proof. (i)⇒(ii): Suppose that 𝐻 is 𝑆-quasinormal in 𝐺. So it follows by [1, Satz 1, page 209] that 𝐻 is subnormal in 𝐺 and then by [10, Lemma 8.6(a), page 28] that 𝐻≤𝑂_𝑝(𝐺). Since 𝐻 is 𝑆-quasinormal in 𝐺, obviously, it is 𝑆-semipermutable in 𝐺. Thus (ii) holds.
(ii)⇒(i): Since 𝐻 is 𝑆-semipermutable in 𝐺, then 𝐻𝐺_𝑞=𝐺_𝑞𝐻 for every Sylow 𝑞-subgroup 𝐺_𝑞 of 𝐺 with (𝑞,|𝐻|)=1. Clearly, 𝐻=𝑂_𝑝(𝐺)∩𝐻𝐺_𝑞 is normal in 𝐻𝐺_𝑞 and so 𝐺_𝑞≤𝑁_𝐺(𝐻). Thus 𝑂^𝑝(𝐺)≤𝑁_𝐺(𝐻). Applying Lemma 2.2, we have that 𝐻 is 𝑆-quasinormal in 𝐺. Thus (i) holdes.

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

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

Lemma 2.6 (see [12, Lemma 2.6]). Let 𝑁 be a nontrivial normal subgroup of a group 𝐺. If 𝑁∩Φ(𝐺)=1, then the Fitting subgroup 𝐹(𝑁) of 𝑁 is the direct product of minimal normal subgroups of 𝐺 which are contained in 𝐹(𝑁).

Lemma 2.7 (see [13, Lemma 3.3.1, page 23]). Suppose that 𝐺_𝑝 is a normal Sylow 𝑝-subgroup of 𝐺 and that Ω(𝐺_𝑝)𝐾 is supersolvable, where 𝐾 is a 𝑝^-Hall subgroup of 𝐺. Then 𝐺 is supersolvable.

3. Main Results

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

Proof. Suppose that the theorem is false, and let 𝐺 be a counterexample of minimal order. We prove the following steps.
(1) If 𝑃≤𝑀<𝐺, Then 𝑀 Is 𝑝-Nilpotent
It is clear to see by Lemma 2.1 that the ALPE-subgroups of 𝑃 are 𝑆-semipermutale in 𝑀, so that 𝑀 satisfies the hypothesis of the theorem. Thus, the minimality of 𝐺 yields that 𝑀 is 𝑝-nilpotent.
(2) 𝑁_𝐺(𝑃) Is 𝑝-Nilpotent
Suppose that 𝑃 is normal in 𝐺. Let 𝐻 be an ALPE-subgroup of 𝑃 (in particular, we may take 𝐻=𝑃). By hypothesis, 𝐻 is 𝑆-semipermutable in 𝐺 and so by Lemma 2.3, we have that 𝐻 is 𝑆-quasinormal in 𝐺. Hence 𝐻𝐺_𝑞 is a subgroup of 𝐺, where 𝐺_𝑞 is a Sylow 𝑞-subgroup of 𝐺 with 𝑞≠𝑝. Clearly, 𝐻 is a subnormal Hall subgroup of 𝐻𝐺_𝑞. Thus 𝐻 is normal in 𝐻𝐺_𝑞 and hence 𝐻 is normal in 𝑃𝐺_𝑞 as 𝑃 is abelian. Thus 𝑃 is supersolvably embedded in 𝑃𝐺_𝑞 by Lemma 2.4 and so 𝑃≤𝑄_∞(𝑃𝐺_𝑞). Since 𝑄_∞(𝑃𝐺_𝑞)≤genz_∞(𝑃𝐺_𝑞) by [14, page 34], it follows by Lemma 2.5 that 𝑃𝐺_𝑞 is 𝑝-nilpotent. Thus 𝑃𝐺_𝑞=𝑃×𝐺_𝑞. Hence 𝐺_𝑞≤𝐶_𝐺(𝑃), so that 𝑂^𝑝(𝐺)≤𝐶_𝐺(𝑃).If𝐶_𝐺(𝑃)<𝐺, then 𝐶_𝐺(𝑃) is 𝑝-nilpotent by (1). Thus 𝑂^𝑝(𝐺) is 𝑝-nilpotent and so 𝐺 is 𝑝-nilpotent: a contradiction. Thus we may assume that 𝐶_𝐺(𝑃)=𝐺. Then 𝑃≤𝑍(𝐺), in particular, 𝑃≤𝑍(𝐺_𝑝). So, 𝑃=𝐺_𝑝 by the maximality of 𝑃 and we have by [15, Theorem 4.3, page 252] that 𝐺 is 𝑝-nilpotent: a contradiction. Thus we may assume that 𝑁_𝐺(𝑃)<𝐺. According to (1), we have that 𝑁_𝐺(𝑃) is 𝑝-nilpotent.
(3) 𝑂_𝑝′(𝐺)=1
If 𝑂_𝑝′(𝐺)≠1, we consider the quotient group 𝐺/𝑂_𝑝′(𝐺). Clearly, 𝐺_𝑝𝑂_𝑝′(𝐺)/𝑂_𝑝′(𝐺) is a Sylow 𝑝-subgroup of 𝐺/𝑂_𝑝′(𝐺) and 𝑃𝑂_𝑝′(𝐺)/𝑂_𝑝′(𝐺) is an ALPE-Subgroup of 𝐺_𝑝𝑂_𝑝′(𝐺)/𝑂_𝑝′(𝐺) having maximal order. By Lemma 2.1, the hypotheses are inherited over 𝐺/𝑂_𝑝′(𝐺). Thus, the minimality of 𝐺 implies that 𝐺/𝑂_𝑝′(𝐺) is 𝑝-nilpotent, hence 𝐺 is 𝑝-nilpotent, which is a contradiction.
(4) 𝐺=𝐺_𝑝𝐺_𝑞, Where 𝐺_𝑞 Is a Sylow 𝑞-Subgroup of 𝐺 with 𝑞≠𝑝
Since 𝐺 is not 𝑝-nilpotent by [15, Theorem 4.5, page 253], there exists a subgroup 𝐻 of 𝐺_𝑝 such that 𝑁_𝐺(𝐻) is not 𝑝-nilpotent. But 𝑁_𝐺(𝐺_𝑝) is 𝑝-nilpotent by a similar argument of the proof of the step (2). Thus we may choose a subgroup 𝐻 of 𝐺_𝑝 such that 𝑁_𝐺(𝐻) is not 𝑝-nilpotent but 𝑁_𝐺(𝐾) is 𝑝-nilpotent for every subgroup 𝐾 of 𝐺_𝑝 with 𝐻<𝐾≤𝐺_𝑝. It is easy to see that 𝑁_𝐺(𝐺_𝑝)≤𝑁_𝐺(𝐻)≤𝐺. If 𝑁_𝐺(𝐻)<𝐺, it follows by (1) that 𝑁_𝐺(𝐻) is 𝑝-nilpotent: a contradiction. Thus 𝑁_𝐺(𝐻)=𝐺. This leads to 𝑂_𝑝(𝐺)≠1 and 𝑁_𝐺(𝐾) is 𝑝-nilpotent for every subgroup 𝐾 of 𝐺_𝑝 with 𝑂_𝑝(𝐺)<𝐾≤𝐺_𝑝. Now, by [15, Theorem 4.5, page 253] again, we see that 𝐺/𝑂_𝑝(𝐺) is 𝑝-nilpotent and therefore that 𝐺 is 𝑝-solvable. Since 𝐺 is 𝑝-solvable, for any 𝑞∈𝜋(𝐺) with 𝑞≠𝑝, there exists a Sylow 𝑞-subgroup 𝐺_𝑞 of 𝐺 such that 𝐺_𝑝𝐺_𝑞≤𝐺 by [15, Theorem 3.5, page 229]. If 𝐺_𝑝𝐺_𝑞<𝐺, then 𝐺_𝑝𝐺_𝑞 is 𝑝-nilpotent by (1) and hence 𝑂_𝑝(𝐺)𝐺_𝑞 is 𝑝-nilpotent. Thus 𝑂_𝑝(𝐺)𝐺_𝑞=𝑂_𝑝(𝐺)×𝐺_𝑞. This leads to 𝐺_𝑞≤𝐶_𝐺(𝑂_𝑝(𝐺))≤𝑂_𝑝(𝐺) by [15, Theorem 3.2, page 228] as 𝑂_𝑝′(𝐺)=1 by (3), which is a contradiction. Thus 𝐺=𝐺_𝑝𝐺_𝑞.
(5) The Final Contradiction
Let 𝑁 be a minimal normal subgroup of 𝐺 such that 𝑁≤𝑂_𝑝(𝐺). Clearly, 𝑁∩𝑍(𝐺_𝑝)≠1 and so 𝑍(𝐺_𝑝)≤𝑃 by the maximality of 𝑃. Hence 1≠𝑁∩𝑍(𝐺_𝑝)≤𝑁∩𝑃. By hypothesis, 𝑃𝐺_𝑞≤𝐺 for any Sylow 𝑞-subgroup 𝐺_𝑞 of 𝐺 with (𝑞,|𝑃|)=1. It is easy to see that 𝑁∩𝑃=𝑁∩𝑃𝐺_𝑞⊲𝑃𝐺_𝑞. Thus 𝑂^𝑝(𝐺)≤𝑁_𝐺(𝑁∩𝑃). If 𝑁_𝐺(𝑁∩𝑃)<𝐺, then by (1), 𝑁_𝐺(𝑁∩𝑃) is 𝑝-nilpotent. Hence 𝑂^𝑝(𝐺) is 𝑝-nilpotent and so also does 𝐺: a contradiction. Thus we may assume that 𝑁_𝐺(𝑁∩𝑃)=𝐺. By the minimality of 𝑁 and since 𝑁∩𝑃≠1, we have that 𝑁∩𝑃=𝑁 and so 𝑁≤𝑃. If 𝑃𝐺_𝑞<𝐺, then 𝑃𝐺_𝑞 is 𝑝-nilpotent by (1) and hence 𝑁𝐺_𝑞 is 𝑝-nilpotent. Thus 𝑁𝐺_𝑞=𝑁×𝐺_𝑞 and so 𝐺_𝑞≤𝐶_𝐺(𝑁). Thus by (4), 𝐺/𝐶_𝐺(𝑁) is a 𝑝-group and so by [14, Theorem 6.3, page 221], 𝑁≤𝑍_∞(𝐺). Since 𝑍_∞(𝐺)≤𝑄_∞(𝐺), we have that 𝑁≤𝑄_∞(𝐺) which implies that 𝑁 is supersolvably embedded in 𝐺 and so clearly that |𝑁|=𝑝. Thus, it is easy to see that the quotient group 𝐺/𝑁 satisfies the hypothesis of the theorem by Lemma 2.1. Now, by the minimality of 𝐺, we see that 𝐺/𝑁 is 𝑝-nilpotent. Since the class of all 𝑝-nilpotent groups is a saturated formation, we have that 𝑁 is the unique minimal normal subgroup of 𝐺 and ^𝑁⩽/^Φ(𝐺). Thus Φ(𝐺)=1 and hence 𝑁=𝑂_𝑝(𝐺) by Lemma 2.6 and so 𝐹(𝐺)=𝑂_𝑝(𝐺)=𝑁 by (3). Hence 𝐺_𝑞≤𝐶_𝐺(𝐹(𝐺)). Since 𝐺 is solvable, it follows by [15, Theorem 2.6, page 216] that 𝐶_𝐺(𝐹(𝐺))≤𝐹(𝐺)=𝑂_𝑝(𝐺): a contradiction. Thus we must have 𝐺=𝑃𝐺_𝑞. Let 𝐺^∗𝑞 be a Sylow 𝑞-subgroup of 𝑁_𝐺(𝑃). By (2), we have that 𝐺^∗𝑞⊲𝑁_𝐺(𝑃). Hence 𝑁_𝐺(𝑃)=𝑃𝐺^∗𝑞=𝑃×𝐺^∗𝑞. Thus 𝑃≤𝑍(𝑁_𝐺(𝑃)), and, therefore, 𝐺 is 𝑝-nilpotent by [15, Theorem 4.3, page 252]: a final contradiction.

We need the following result.

Theorem 3.2. Let ℱ be a saturated formation containing the class of supersolvable groups 𝒰. Let 𝐺_𝑝 be a normal Sylow 𝑝-subgroup of a group 𝐺 such that 𝐺/𝐺_𝑝∈ℱ. Fix an ALPE-subgroup 𝑃 of 𝐺_𝑝 having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺∈ℱ.

Proof. We treat the following two cases.
Case 1. 𝑂_𝑝′(𝐺)≠1.
Clearly, 𝐺_𝑝𝑂_𝑝′(𝐺)/𝑂_𝑝′(𝐺) is a normal Sylow 𝑝-subgroup of 𝐺/𝑂_𝑝′(𝐺) and 𝑃𝑂_𝑝′(𝐺)/𝑂_𝑝′(𝐺) is an ALPE-subgroup of 𝐺_𝑝𝑂_𝑝′(𝐺)/𝑂_𝑝′(𝐺) having maximal order. By hypothesis and Lemma 2.1, the ALPE-subgroups of 𝑃𝑂_𝑝′(𝐺)/𝑂_𝑝′(𝐺) are 𝑆-semipermutable in 𝐺/𝑂_𝑝′(𝐺). Clearly, 𝐺/𝐺_𝑝𝐺_𝑝𝑂_𝑝′(𝐺)/𝐺_𝑝≅𝐺_𝑝𝑂𝐺^𝑝′(𝐺)≅𝐺/𝑂^𝑝′(𝐺)𝐺_𝑝𝑂_𝑝′(𝐺)/𝑂_𝑝′(𝐺)∈ℱ.(3.1)
Thus, our hypothesis carries over to 𝐺/𝑂_𝑝′(𝐺) and so 𝐺/𝑂_𝑝′(𝐺)∈ℱ by induction on the order of 𝐺. Therefore, 𝐺/(𝑂_𝑝′(𝐺)∩𝐺_𝑝)≅𝐺∈ℱ.
Case 2. 𝑂_𝑝′(𝐺)=1.
Let 𝐻 be an ALPE-subgroup of 𝑃. Then 𝐻 is 𝑆-quasinormal in 𝐺 by Lemma 2.3 and hence 𝑂^𝑝(𝐺)≤𝑁_𝐺(𝐻) by Lemma 2.2. Let 𝑇=𝑃𝑂^𝑝(𝐺). Then 𝐻 is normal in 𝑇. Thus Lemma 2.4 implies that 𝑃 is supersolvably embedded in 𝑇. Then, 𝑇/𝐶_𝑇(𝑃) is supersolvable by [14, Lemma 7.15, page 35]. Clearly, 𝑇_𝑝=𝐺_𝑝∩𝑇⊲𝑇, where 𝑇_𝑝 is a Sylow 𝑝-subgroup of 𝑇. Let 𝑄 be a 𝑝^-subgroup of 𝐶_𝑇(𝑃). Then 𝑄𝑃=𝑄×𝑃 is a group of automorphisms of 𝑇_𝑝=𝑂_𝑝(𝑇). But 𝐶_𝑇𝑝(𝑃)=𝑃, and consequently, 𝑄 acts trivially on 𝐶_𝑇𝑝(𝑃). Then 𝑄 acts trivially on 𝑇_𝑝 by [15, Theorem 3.4, page 179], that is, 𝑄≤𝐶_𝑇(𝑇_𝑝). It is easy to see that 𝑇 is subnormal in 𝐺 and so 𝑂_𝑝′(𝑇)≤𝑂_𝑝′(𝐺)=1.  Hence 𝐹(𝑇)=𝑇_𝑝.  Since 𝑇 is solvable, it follows by [15, Theorem 2.6, page 216] that 𝑄≤𝐶_𝑇(𝐹(𝑇))≤𝐹(𝑇)=𝑇_𝑝: a contradiction. Hence 𝐶_𝑇(𝑃) must be a 𝑝-group and so 𝐶_𝑇(𝑃)=𝑃. Thus, 𝑇/𝐶_𝑇(𝑃)=𝑇/𝑃 is supersolvable which implies that 𝑇 is supersolvable by [16, Theorem 4]. Thus 𝑂^𝑝(𝐺) is supersolvable and therefore, 𝐺=𝐺_𝑝𝑂^𝑝(𝐺) is supersolvable by [17, Exercise 7.2.23, page 159]. Hence, 𝐺∈𝒰⊆ℱ.

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

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

We now prove the following theorem.

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

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

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

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

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

We need the following Lemma.

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

Proof. Let 𝐺_𝑝 be a Sylow 𝑝-subgroup of 𝐺. We treat the following two cases.
Case 1. 𝐾=𝐺_𝑝.
Then by Corollary 3.3, 𝐺 is supersolvable.
Case 2. 𝐾<𝐺_𝑝.
Put 𝜋(𝐺)={𝑝₁,𝑝₂,…,𝑝_𝑛}, where 𝑝₁>𝑝₂>⋯>𝑝_𝑛. Since 𝐺/𝐾 is supersolvable, it follows by [18, Theorem 5, page 5] that 𝐺/𝐾 possesses supersolvable subgroups 𝑀/𝐾 and 𝐿/𝐾 such that |𝐺/𝐾∶𝑀/𝐾|=𝑝₁ and |𝐺/𝐾∶𝐿/𝐾|=𝑝_𝑛. Since 𝑀/𝐾 and 𝐿/𝐾 are supersolvable, it follows that 𝑀 and 𝐿 are supersolvable by induction on the order of 𝐺. Since |𝐺∶𝑀|=|𝐺/𝐾∶𝑀/𝐾|=𝑝₁ and |𝐺∶𝐿|=|𝐺/𝐾∶𝐿/𝐾|=𝑝_𝑛, it follows again by [18, Theorem 5, page 5] that 𝐺 is supersolvable.

Now, we can prove the following theorem.

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

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

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

Corollary 3.9 (Ramadan [8]). Let 𝐾 be a normal subgroup of a group 𝐺 such that 𝐺/𝐾 is supersolvable. If the ALPE-subgroups of every Sylow subgroup of 𝐾 are 𝑆-quasinormal in 𝐺, then 𝐺 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 𝑝 be the smallest prime dividing the order of a group 𝐺 and let 𝐺_𝑝 be a Sylow 𝑝-subgroup of 𝐺. Fix an ALPE-subgroup 𝑃 of Ω(𝐺_𝑝) having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is 𝑝-nilpotent.

We prove the following lemma.

Lemma 4.2. Let 𝐾 be a normal 𝑝-subgroup of a group 𝐺 such that 𝐺/𝐾 is supersolvable. Fix an ALPE-subgroup 𝑃 of Ω(𝐾) having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

Proof. Let 𝐺_𝑝 be a Sylow 𝑝-subgroup of 𝐺. We treat the following two cases.
Case 1 𝐾=𝐺_𝑝. By [15, Theorem 2.1, page 221], there exists a 𝑝^-Hall subgroup 𝑇, which is a complement to 𝐺_𝑝 in 𝐺. Hence 𝐺/𝐺_𝑝≅𝑇 is supersolvable. Since Ω(𝐺_𝑝) is characteristic in 𝐺_𝑝 and 𝐺_𝑝⊲𝐺, we have that Ω(𝐺_𝑝)⊲𝐺. Clearly, Ω(𝐺_𝑝)𝑇/Ω(𝐺_𝑝)≅𝑇 is supersolvable. Thus, our hypothesis and Corollary 3.3 imply that Ω(𝐺_𝑝)𝑇 is supersolvable. Therefore, 𝐺 is supersolvable by Lemma 2.7.
Case 2 𝐾<𝐺. Put 𝜋(𝐺)={𝑝₁,𝑝₂,…,𝑝_𝑛}, where 𝑝₁>𝑝₂>⋯>𝑝_𝑛. Since 𝐺/𝐾 is supersolvable, it follows by [18, Theorem 5, page 5] that 𝐺/𝐾 possesses supersolvable subgroups 𝑀/𝐾 and 𝐿/𝐾 such that |𝐺/𝐾∶𝑀/𝐾|=𝑝₁ and |𝐺/𝐾∶𝐿/𝐾|=𝑝_𝑛. Since 𝑀/𝐾 and 𝐿/𝐾 are supersolvable, it follows that 𝑀 and 𝐿 are supersolvable by induction on the order of 𝐺. Since |𝐺∶𝑀|=|𝐺/𝐾∶𝑀/𝐾|=𝑝₁ and |𝐺∶𝐿|=|𝐺/𝐾∶𝐿/𝐾|=𝑝_𝑛, it follows again by [18, Theorem 5, page 5] that 𝐺 is supersolvable.

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

Theorem 4.3. Let 𝐺 be a group. For every prime 𝑝 and every Sylow 𝑝-subgroup 𝐺_𝑝 of 𝐺,fix an ALPE-subgroup 𝑃 of Ω(𝐺_𝑝) having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

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

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

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

We can now prove the following corollary.

Corollary 4.6. Let 𝐾 be a normal subgroup of 𝐺 such that 𝐺/𝐾 is supersolvable. For every prime 𝑝 dividing the order of 𝐾 and every Sylow 𝑝-subgroup 𝐾_𝑝 of 𝐾, fix an ALPE-subgroup 𝑃 of Ω(𝐾_𝑝) having maximal order. If the ALPE-subgroups of 𝑃 are 𝑆-semipermutable in 𝐺, then 𝐺 is supersolvable.

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

Remarks 4.7. (a) The converse of Theorem 3.4 is not true. For example, set 𝐺=𝑆₃×𝑍₃, where 𝑆₃=⟨𝑥,𝑦∣𝑥³=𝑦²=1,𝑦𝑥=𝑥²𝑦⟩ and 𝑍₃=⟨𝑧∣𝑧³=1⟩. Clearly, 𝐺 is supersolvable and 𝐺 has an abelian Sylow 3-subgroup of exponent 3. It is easy to check that 𝐺 contains a subgroup ⟨𝑥𝑧⟩ of order 3 which fails to be 𝑆-semipermutable in 𝐺.
(b) Theorem 4.3 is not true when the smallest prime dividing the order of 𝐺 is even and Ω(𝐺_𝑝)=Ω₁(𝐺_𝑝), where 𝐺_𝑝 is a Sylow 𝑝-subgroup of 𝐺. For example, if 𝑄 is the quaternion group ⟨𝑎,𝑏∣𝑎⁴=1,𝑏²=𝑎²,𝑏⁻¹𝑎𝑏=𝑎⁻¹⟩, 𝐶₉ is a cyclic group of order 9 with generator 𝑐, and the action of 𝐶₉ on 𝑄 is given by 𝑎^𝑐=𝑏,𝑏^𝑐=𝑎𝑏, then the semidirect product of 𝑄 by 𝐶₉ is a group of even order in which every subgroup of prime order is 𝑆-semipermutable. Clearly, the semidirect product of 𝑄 by 𝐶₉ is not supersolvable (see Buckley [4, Examples (ii)]).