EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6033 ISSN 1307-5543 – ejpam.com Published by New York Business Global Notes on Finite Groups with Nearly Sylow-permutable and Nearly Sylow-permutable-Transitive Subgroups Abdulaziz Mutlaq Alotaibi1, Khalid Al-Tahat2, Khaled Mustafa Al-Jamal2,∗ 1 Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Saudi Arabia 2 Faculty of Computer Studies, Arab Open University, Amman, Jordan Abstract. Let G be a finite group and let H be a subgroup of G. We called H is nearly S- permutable in G if for every prime p such that (p: |H |= 1) p-subgroup of K. We shall denote this by (H is NSP in G). We introduce the class of nearly S-permutable transitive -groups as those groups in which nearly S -permutability is transitive among subgroups. That is, if A is NSP in B ; and B is NSP in G, then A is NSP in G : In this paper we study some characterize finite groups using NSP and NSPT and we compare some subgroups with groups under study, supported by theorems and examples. 2020 Mathematics Subject Classifications: 20D10, 20D20, 20D35 Key Words and Phrases: S-permutable subgroup, Sylow subgroup, permutable subgroup, nearly S-permutable subgroup, nearly S-permutable 1. Introduction In this paper, all groups under discussion are finite. A subgroup H of a group G is said to commute with another subgroup K if the product HK is also a subgroup of G. If H commutes with every subgroup (or every Sylow subgroup) of G, it is called a permutable (or S-permutable) subgroup [1]. A well-known fact in group theory is that normal p-Sylow subgroups imply nilpotency, as normality plays a central role in subgroup structure. One fundamental property of normal subgroups is that if N is a normal subgroup of G and H is any subgroup of G, then NH = HN . However, it has been observed that some subgroups, although not normal, still com- mute with every subgroup in the group [2]. According to [1], an S-permutable subgroup of a group is subnormal. In contrast, nearly S-permutability does not necessarily imply subnormality [3]. A notable example is the dihedral group D18 = ⟨r, s : r9 = s2 = e, rs = sr8 ⟩, which contains nearly S-permutable subgroups that are not subnormal [4, 5]. This ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6033 Email addresses: am.alotaibi@psau.edu.sa (A. M. Alotaibi), ktahat@aou.edu.jo (K.Al − Tahat), pt aljammal@aou.edu.jo (K.M.Al − Jamal) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. M. Alotaibiang, K. Al-Tahat, K. M. Al-Jamal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6033 2 of 8 highlights the significance of studying the structural differences between these subgroup properties. Motivated by this, our research focuses on nearly S-permutable subgroups and in- troduces a new class of groups called NSPT -groups, in which the property of nearly S-permutability is transitive among subgroups. That is, if A is nearly S-permutable in B, and B is nearly S-permutable in G, then A is nearly S-permutable in G [6]. Several generalizations of normality and subnormality have been investigated in the literature, including c-normality and S-permutability. For example, soluble T -groups (where every subnormal subgroup is normal) were studied in [7], and similar results for S- permutability were developed in [8, 9]. Analogous results for nearly S-permutability were given in [10], where the structure of soluble groups with nearly S-permutable subnormal subgroups was described. 2. Preliminaries This section will introduce fundamental theories and concepts related to groups and subgroups, which serve as the foundation for abstract algebra and will be applied later. Definition 1. [10] A subgroup H of G is termed nearly S-permutable in G if, for every prime p that does not divide the order of H, and for every subgroup K of G containing H, the normalizer NK(H)includes at least one Sylow p-subgroup of K. We use the notation H nspG to signify that H is nearly S-permutable in G. Definition 2. [11] A group G is referred to as an NSPT -group if the property of nearly S-permutability is transitive within G. Specifically, G is an NSPT -group if, for any sub- groups H and K of G such that His nearly S-permutable in K and K is nearly S-permutable in G, it follows that H is nearly S-permutable in G. Lemma 1. [12] Let N ⊴G and suppose that P ∈ Sylp(N), then G = NG(P )N . Definition 3. [8] A subgroup H of G is said to be S-permutable in G if HP = PH holds for every Sylow p-subgroup of G and for every prime p in the set of prime divisors of the order of G, denoted by σ(G). Proposition 1. [13] Let G be a group. Then the following properties hold: (i) If H is normal in G, then H is c-normal in G. (ii) The group G is c-simple if and only if G is simple. (iii) If H is c-normal in G and H ≤ K ≤ G, then H is c-normal in K. (iv) Let K be a normal subgroup of G such that K ≤ H. Then H is c-normal in G if and only if H/K is c-normal in G/K. Definition 4. [9] A T -group is a group where normality is a transitive property, meaning that every subnormal subgroup is normal. Specifically, if H ⊴K and K ⊴G, then H ⊴G. A. M. Alotaibiang, K. Al-Tahat, K. M. Al-Jamal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6033 3 of 8 Lemma 2. Every normal subgroup is nearly S-permutability subgroup. Proof. See [11] Examples of T -groups include abelian groups, Dedekind groups, and simple groups. Definition 5. [14] Let G be a group we called a CT -group if the property of c-normality is transitive in G. Specifically, G is a CT -group if for all subgroups H and K of G, whenever H is c-normal in K and K is c-normal in G, it follows that H is c-normal in G. Corollary 1. [15] All maximal subgroups of a solvable CT -group is a CT -group. 3. Main Results Nilpotent groups can be viewed as an extension of the concept of P -groups. This section explores finite groups where the property of nearly S-permutability is transitive, some re- sults and theorems of finite solvable NSPT -groups were proven. It is clear that all abelian groups and all nilpotent groups are examples of groups with nearly S-permutable groups. But not all groups satisfies this property as the following example shows: Example 1. The alternating group on 4-letters A4 does not satisfiy the nearly S-permutable. Specifically, any of the Sylow 3-subgroups in A4 will not be a nearly S-permutable in A4. Remark 1. Transitive realation of nearly S-permutable is not true for all groups. Lemma 3. Every normal subgroup is nearly S-permutable. Proof. Let G be agroup and H is normal subgroup of G and let H ≤ K ≤ G for every prime number p ∈ P with (P, |H|), since H ⊴ G implies H ⊴ K. Then NK(H) = K.If P ∈ Sylp(K) that implies P ≤ K = NK(H), that mean H is nearly S-permutable.The proof is complete. Proposition 2. The intersection of two nearly S-permutable subgroups does not nec- essarily imply that the intersection is nearly S-permutable, and a subgroup being nearly S-permutable does not imply that all of its subgroups are nearly S-permutable. Proof. Let G be a group of order 18 defined as the direct product of the symmetric group S3 and the cyclic group Z3, i.e., G = S3 × Z3. G = {(e, 0), ((12), 0), ((13), 0), ((23), 0), ((132), 0)((123), 0), (e, 1), ((12), 1), ((13), 1), ((23), 1), ((132), 1), ((123), 1), (e, 2), ((12), 2), ((13), 2), ((23), 2), ((132), 2), ((123), 2)}. The normal subgroups in G are: order 9 : ⟨((132), 0), (e, 1)⟩. order 6 : ⟨((12), 0), ((13), 0)⟩. order 3 : ⟨(e, 1)⟩ and order 1 In this case, every normal subgroup is nearly S-permutable. A. M. Alotaibiang, K. Al-Tahat, K. M. Al-Jamal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6033 4 of 8 However, the intersection of two nearly S-permutable subgroups does not necessar- ily maintain the nearly S-permutable property. Moreover, a subgroup being nearly S- permutable does not guarantee that all its subgroups will also be nearly S-permutable. For the subgroups of order 6, (non-normal subgroup) there are 3 conjugacy classes are nearly S-permutable: H1 = ⟨(13), 1)⟩. H2 = ⟨(23), 1)⟩. H3 = ⟨(12, 1)⟩. Now in this group G we have 9 subgroups that are nearly S-permutable. Take H1 with N a normal subgroup of order 6, we have H1 ∩N = ⟨(12), 0)⟩, now H1 is NSP and N is NSP but the intersection doesnt nearly S-permutable. Now takeH2,the subgroup of H2 is⟨(23, 0)⟩ ,its clear that H2 is NSP. But the subgroup from H2 is not NSP. The proof is complete. Lemma 4. If H is a p-subgroup of G, then H is contained in some Sylow a p-subgroup of G. Proof. Let L = Sylp(G), and H be p-subgroup of G. Consider the action H × L → L, h(p) = h−1ph, then L is a G-set so |L|∼= |LH |(modp). But|L| = |Sylp(G)| =ηp ≡ 1(mod p). . . (1) Let us examine LH . P ∈ LH if and only if hp = p ∀h ∈ H and h−1ph∀h ∈ H.IfH ≤ NG(P ), from (1) there exist at least one P ∈ L, such that H ≤ NG(P ), then HP = PH, it can be seen that HP is a subgroup of NG(P ). Note HP is Sylow p-subgroup NG(P ). Hence,HP = P ; thus, H ≤ P . Remark 2. There exists a CT -group which is not an NSPT -group. Proof. Let G be a finite group of order 18, given by G = S3 ×Z3 . The elements of G are: G = {(e, 0), ((12), 0), ((13), 0), ((23), 0), ((13), 2), ((23), 2), ((132), 2), ((123), 2)}. Normal subgroups in G are: order 9 : ⟨((132), 0), (e, 1)⟩. order 6 : ⟨((12), 0), ((13), 0)⟩. order 3 : ⟨(e, 1)⟩ and order1. All normal subgroups of G satisfy the c-normality condition. Additionally, subgroups with order 6 are distributed across 3 conjugacy classes: H1 = {(e, 0), (e, 1), (e, 2), ((23), 0), ((23), 1), ((23), 2)}. H2 = {(e, 0), (e, 1), (e, 2), ((13), 0), ((13), 1), ((13), 2). H3 = {(e, 0), (e, 1), (e, 2), ((12), 0), ((12), 1), ((12), 2)}. For each Hi, we find that: If H1 is paired with N , a normal subgroup of order 9, then G = H1N and H1 ∩N = ⟨(e, 1)⟩. This ensures H1 ∩N ⊆ (H1), making subgroups of order 6 are c-normal. For subgroups of order 3, there are two distinct conjugacy classes: H1 = {(e, 0), ((123), 1), ((132), 2}. A. M. Alotaibiang, K. Al-Tahat, K. M. Al-Jamal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6033 5 of 8 H2 = {(e, 0), ((132), 1), ((123), 2}. Similarly, when H1 is paired with N , a normal subgroup of order 6, we have G = H1N and H1 ∩ N = ⟨(e, 0)⟩H1 ∩ N = ⟨(e, 0)⟩. This guarantees that subgroups of order 3 are c-normal. For subgroups of order 2, there are 3 conjugacy classes: H1 = {(e, 0), ((23), 0)}. H2 = {(e, 0), ((13), 0)}. H3 = {(e, 0), ((12), 0)}. Subgroups of order 2 are c-normal, as demonstrated by similar reasoning. Now the sylow 3-subgroup of G is ⟨((132), (e, 1)⟩ ≃ Z3 × A3 = P , the indexed [G : P ] = 2. Now p is normal in G, such that NG(H) = G; take H of order 3. Now H is nearly S-permutable, since NG(H) = P and P is nearly S-permutable in G, but H is not nearly S-permutable in G, Hence, G is not NSPT -group. Thus, G is not an NSPT -group, demonstrating that a CT -group need not be an NSPT -group. Remark 3. There exists a finite group which is an NSPT -group but not a CT -group. Proof. Let E8 = Z2 × Z2 × Z2, (E: elementary 2-group of order 8). Now Aut (E8) ≃ PSL(3, 2)-Simple group of order 168 and it has 179 subgroups, 35-subgroups of which are of order 4. If H ≃ Z4 ≤ Aut(E8) = PSL(3, 2) and G = E8 ⋊ H(Z2 × Z2 × Z2) ⋊ Z4, then G is example of an NSPT -group that is not a CT -group. G is not CT -group, G = E8×Z4. More specifically, G = (Z2×Z2×Z2)⋊H ·H = ⟨d⟩, note H is a c-normal in G, since there exist K ≤ G;K ⊴ G and k ∩ h ≤ HG = 1 and HK = G; since, |H| = 4every subgroup of H is normal. But H1 = ⟨d2⟩⊴H, c-normal in G, H1{1, d2} ≃ Z2 and note H1 ≤ M for any subgroup M of G with |M | = 16 implies M ⊴G, there are 3 such M.MG = 1-identity. If KH1 = G with K ⊴ G implies |K| = 16, but H ≤ K for any K. NSPT -group follows from |G| = 25 and G is p-group that implies, every subgroup is nearly S-permutable. From Remark 2 and Remark 3, we conclude that the classes of CT -groups and NSPT - groups are incomparable. That is, a CT -group need not beNSPT , and an NSPT -group need not be CT -groups. Lemma 5. Let G be a finite group and let P be a Sylow p-subgroup of G, it follows that NG(NG(P ) = NG(P ). Proof. Since P ⊆ NG(P ) ⊆ G and P is a Sylow p-subgroup of G therefore, P is a Sylow p-subgroup of NG(P ). Furthermore, P is normal in NG(P ), making P is unique Sylow p-subgroup of G. Now let a ∈ NG(NG(P )). A. M. Alotaibiang, K. Al-Tahat, K. M. Al-Jamal / Eur. J. Pure Appl. Math, 18 (3) (2025), 6033 6 of 8 We will aim show that a ∈ NG(P ). Observe that: aPa −1 ⊆ aNG (P )a−1 = NG(P ). This implies that, aPa−1 is a Sylow p-subgroup of NG(P ), we have aPa−1 = P . This means a ∈ NG(P ). Theorem 1. Let G is a nilpotent group and let H subgroup of G. Then H ≤ NG(H). Proof. Since G is nilpotent, it has a central series{Ni | 0 < i < r}, and we have N0 = 1 ⊆ H and Nr = G ⊈ H. Consequently, there exists an index k with 0 < k < r, such that Nk ⊆ H, butNk+1 ⊊ H. We will show that in fact,Nk+1 ⊆ NG(H), and it will follow that H ≤ NG(H), as required. Theorem 2. Let G be a finite group. Then the following are equivalent. (i) G is nilpotent. (ii) H ≤ NG(H) for every subgroup H ≤ G. (iii) All maximal subgroups of G are nearly S-permutable. Proof. We saw that (i) implies (ii) in Theorem 1 That (ii) implies (iii) is clear, since every maximal subgroup M in G satisfies NG(M) ≥ M . it follows that NG(M) = G. Now assume condition (iii) holds and let P ∈ Sylp(G) for some prime p. If NG(P ) is proper in G, it must be contained in some maximal subgroup M , and we have M ≤ G. Since P ∈ Sylp(M), it follows by Lemma 1 and Lemma 5, that G = NG(P )M ⊆ M , and this is a contradiction. Thus P is normal in G and by Lemma 3, P is nearly S-permutable. Example 2. Let G = Z2×Z2×Z2, an abelian group of order 8. Then: G is nilpotent (since all abelian groups are nilpotent). Every subgroup H ≤ G satisfies H ≤ NG(H) = G. Every maximal subgroup is of order 4 and is normal in G, and hence nearly S-permutable. This example confirms that all three conditions hold simultaneously, illustrating the equivalence. 4. Conclusion New facts on Sylow p-subgroups and nearly S-permutable subgroups were discovered and established. The relationship between CT -groups and NSPT -groups was clarified. It was demonstrated that every normal subgroup is nearly S-permutable, whereas a subnormal subgroup is not necessarily nearly S-permutable. Moreover, a new class of groups, termed NSPT -groups, was introduced, characterized by the transitivity of nearly S-permutability. Additionally, it was proven that c-normal subgroups do not necessarily exhibit nearly S- permutability, and conversely, nearly S-permutability does not imply c-normality. Future research may focus on the following directions: 1. Investigating relationships between NSPT -groups and other group classes with similar subgroup properties. 2. Constructing new group classes inspired by NSPT -groups and comparing them with nilpotent and solvable groups. A. M. Alotaibiang, K. Al-Tahat, K. M. Al-Jamal / Eur. J. Pure Appl. 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