EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 2, 2023, 1118-1127 ISSN 1307-5543 – ejpam.com Published by New York Business Global Nonabelian Case of Hopf Galois Structures on Nonnormal Extensions of Degree pqw Baraa M. Jamal1, Ali A. Alabdali1,∗ 1 Department of Mathematics, College of Education for Pure Science, University of Mosul, Mosul, Iraq Abstract. We look at Hopf Galois structures with square free pqw degree on separable field extensions (nonnormal) L/K. Where E/K is the normal closure of L/K, the group permutation of degree pqw is G = Gal(E/K). We study details of the nonabelian case, where Jl = ⟨σ, [τ, αl]⟩ is a nonabelian regular subgroup of Hol(N) for 1 ≤ l ≤ w − 1. We first find the group permutation G, and then the Hopf Galois structures for each G. In this case, there exists four G such that the Hopf Galois structures are admissible within the field extensions L/K. 2020 Mathematics Subject Classifications: 12F10, 16T05 Key Words and Phrases: Hopf Galois structures, field extensions, groups of square free order, Sophie Germain primes 1. Introduction Chase and Sweedler [7] proposed the Hopf Galois theory by investigating inseparable field extensions. Their work marks the start of a slew of new problems about separable field extensions (SFEs). In [14] Greither and Pareigis showed that an SFE can generate a large number of Hopf Galois structures (HGSs), and HGSs can be used by the group theoretic in issues. If the field extension L/K is normal and separable with degree n, then the Galois extension L/K is classical Galois. Let its Galois group be G = Gal(L/K). The group algebra K[G] then operates on L/K, yielding at least one HGS. On the other hand, there could be a slew of more HGSs on L/K. We have L as Hopf algebras L⊗K H ∼= L[N ] for every group N of order n if the K Hopf algebra H generates one of these HGSs on L/K. We have the type of HGS by the isomorphism type of the group N . The group G determinates the different types of HGS as well as the number of each type. Consider L/K to be an SFE (presumably nonnormal) of degree n in general. Let the nor- mal closure of L/K is F/K, while the Galois groups of F/K and F/L are G = Gal(F/K) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4755 Email addresses: baraa.20esp7@student.uomosul.edu.iq (B. Jamal), ali.alabdali@uomosul.edu.iq (A. Alabdali) https://www.ejpam.com 1118 © 2023 EJPAM All rights reserved. B. Jamal, A. Alabdali / Eur. J. Pure Appl. Math, 16 (2) (2023), 1118-1127 1119 and G ′ = Gal(F/L), respectively. In each type the number of HGSs is determined by the group G and its subgroup G ′ . The primary finding of Greither and Pareigis [14] is that HGSs on L/K are congruent to order n groups and act transitively as a permutation group (PG) on the space of left coset X = G/G ′ . Many authors have studied HGSs since Greither and Pareigis’ efforts on the subject. The majority of them are interested in Galois extensions on various forms of SFEs; see [11, 16, 19]. Other authors, such as [9, 12, 13] deal with nonnormal extensions. The study of HGSs on Galois extensions has been more well-known in recent years due to a link between studying HGSs and the solutions of the Yang Baxter equation as set theoretic (skew braces and braces), for more information, see [2, 18]. Byott shows in [4] that there exists a CG of order pq and a nonabelian group of the same order such that a Galois extension of degree pq allows the number of HGSs, based on the condition p ≡ 1 (mod q) where p and q are distinct primes. A Galois extension with kinds of groups admits the cyclic and nonabelian HGSs. Furthermore, when the degree is 2pq with odd primes p, q and p = 2q + 1 (p is safe prime and q is Sophie Germain prime), there is research in numerous resources [5, 10, 17]. The HGSs on L/K of type N with an arbitrary square free degree n are described in [1]. The HGSs were enumerated by dividing the order n into two groups, G and N , with G = Gal(L/K). Byott and Lyons has shown in their paper [6] that the conclusions of [1] may extend to nonnormal but SFEs L/K of square free degree n = pq (p = 2q + 1 is a safe prime and q ≥ 3 is a Sophie Germain prime). There is at least one cyclic and nonabelian HGS for the PGs admitted by the corresponding field extensions L/K. The issue in [6] then becomes whether the same behaviour applies for square free degrees n in general. The primary purpose of this research is to answer the question and extend the approach in [6] for n = pqw, where a Sophie Germain prime w ≥ 3, a safe prime p = 2w+1, and p, q, w are square free primes. We start with the potential group Jl of order pqw in this work, and then look for PGs that are released by HGSs of type Jl. Where Jl is the nonabelian group. We then enumerate all HGSs of cyclic type on Jl -extension and identify all isomorphism types of PGs of degree pqw. Now we can show the first of our main results. Theorem 1. The total number of isomorphism types admits nonabelian HGSs is 2+ (r+ 1) + σ0(s) of PGs of degree pqw. The nonabelain of order pqw is the regular group. The second of our main results shows the total isomorphism types that realise by cyclic and nonabelian group. Theorem 2. A HGS of cyclic type can realise isomorphism types in total 12(r + i + 1)[σ0(s) + σ1(j) + σ0(s)σ1(j)] + 2 + (r + 1) + σ0(s) of PGs G of degree pqw of both cases regular groups (where the Galois extensions have 1 HGS for the cyclic group and (p − 1)(q − 1) + 1, 1, r + 1, and σ0(s) HGSs for the nonabelian group of the cyclic type). B. Jamal, A. Alabdali / Eur. J. Pure Appl. Math, 16 (2) (2023), 1118-1127 1120 2. Materials and Methods In this section, we review the fundamental facts and concepts related to HGSs, as well as the relationship between them and PGs. Using the method provided in [3], we demonstrate how to count HGSs. We recommend [ [8], Chapter 2] to the reader for further information on counting HGSs. A PG is defined as a finite group G with a one to one homomorphism ρ from G into the PG of a finite set X (ρ : G→ Perm(X)). let y ∈ X,h ∈ G then we express ρ(h)(x) = h.y. The degree of G is the order of X. If there is a unique h ∈ G (respectively, some h ∈ G) with h.y = x for each x, y ∈ X, then G is regular (respectively, transitive) on X. We assume all PGs to be transitive groups in our study. We define the subgroup Gy = {h ∈ G : h.y = y} as the stabilizer of y ∈ X, so the stabilizer of h.y is referred to be as hGyh −1. The core ∩h∈GhGyh −1 of Gy in G is simple as a result of the fact that X affects transitively by G and G embedded in Perm(X). Additionally, by the left multiplication action µ : G → Perm(G/Gy), G acts as a PG on G/Gy = {hGy : h ∈ G} the set of left cosets, where µ(h)(h ′ Gy) = (hh ′ )Gy. As a result, the left translation on G/G ′ acts up to isomorphism on the abstract group G as a PG of degree n, with the subgroup G ′ having a trivial core with index n. We utilize the automorphism definition. If Aut(G,G ′ ) is defined as Aut(G,G ′ ) = {ϕ ∈ Aut(G) : ϕ(G ′ ) = G ′}. Thus it is clear that Aut(G,G ′ ) forms a PG of automorphisms ϕ of G such that ϕ fixes the left coset 1GG ′ of G/G ′ (1G is the identity of G). Let we have a finite SFE L/K of degree n with a fixed algebraic closure F as normal closure in Kc of K. If the group G = Gal(F/K) and the group G ′ = Gal(F/L), then the map µ : G → Perm(X) is an embedding. G ′ acts as a stabilizer for the inclusion L ↪→ F , and the embeddings of K linear of L into Kc or F is acted transitively by G. Consider the cocommutative K Hopf algebra H. Let ϵ : H → K be the counit map for K and ν : H → H ⊗K H be the comultiplication map for ν(ξ) = ∑ (ξ) ξ(1) ⊗ ξ(2). If we have ξ(ab) = ∑ (ξ) ξ(1)(a)ξ(2)(b) for ξ ∈ H and a, b ∈ L, and ξ(k) = ϵ(ξ)k for all ξ ∈ H and k ∈ K, L is said to have H module algebra. In addition, if ϕ : L ⊗K H → EndK(L) described as the K module homomorphism by ϕ(a ⊗ ξ)(b) = aξ(b) is an isomorphism, we say that L/K is a H Galois extension or that H yields an HGS on L/K. The PG G is necessary to obtain the HGSs on L/K. Greither and Pareigis’ discovery is that the left translation group µ(G) normalizes the regular subgroups N of Perm(X) that are isomorphic to the HGSs on L/K. The Hopf algebra of K for each such group N , H = F [N ]G acts on L via Galois descent, where F [G] is acted by G as an automorphisms field of F and conjugates on N via µ. The HGS type is also known as the N isomorphism type. If N = C, where C is the normal complement of the subgroup G ′ of G, we get an HGS. The classical HGSs on L/K is then admitted by F [N ]G. B. Jamal, A. Alabdali / Eur. J. Pure Appl. Math, 16 (2) (2023), 1118-1127 1121 If the isomorphism ϕ : G → Gal(F/K) with ϕ(G ′ ) = Gal(F/L) exists, we have G is realised by an SFE L/K. The point stabilizer is G ′ , and the PG is G. In addition, we state that type N of an HGS realises G if it is admitted by L/K. We count the type N of the number of HGSs as a group of order n such that G is realised by a SFE L/K, which corresponds to the number of regular subgroups N∗ normalised by µ(G) and isomorphic to N of Perm(X), according to Greither and Pareigis conclusions. Let Hol(N) = N ⋊Aut(N) be the holomorph of N . As a result, the number of HGSs on L/K can be determined using Byott’s result in [3] and the formula f(G,N) = |Aut(G,G′ )| |Aut(N)| f ′ (G,N), (1) we denote f ′ (G,N) as the number of regular subgroups B with transitive on N of Hol(N) and B ∼= G by an isomorphism with the stabilizer B ′ of 1N in B to G ′ . If HGS of type N realises G, then G ∼= B of Hol(N). Because the preceding approach deals with Hol(N) instead of the Perm(X) group, count- ing HGSs is made easy. We write the elements of Hol(N) by [x, α] where x ∈ N and α ∈ Aut(N). Thus Hol(N) acts on N as permutations by [x, α].y = xα(y). Then, in Hol(N) the normal subgroup N specifies µ(N) of the left translations, and the stabilizer of 1N creates the subgroup Aut(N). In Hol(N), the multiplication is defined as follows [x, α][y, β] = [xα(y), αβ]. We commonly refer to x and α instead of [x, iN ] and [1N , α] the elements of Hol(N), respectively. For example, we have the identification αx = α(x)α. Now we have some general results from [6] about holomorphs, the group N and Aut(N). Proposition 1. Assume that N and Aut(N) are abelian group and abelian automorphism respectively. Suppose the two subgroups of Hol(N), B = N⋊A and B ′ = N⋊A′ such that A,A ′ two subgroups of Aut(N). Let ψ : B → B ′ be an isomorphism such that ψ(N) = N , therefore B = B ′ . Proposition 2. Suppose that N is a group such that A is a subgroup of Aut(N). Suppose that the subgroup of Hol(N) is B = N ⋊ A such that N is characteristic in B. Then the normalizer of A in automorphism of N is isomorphic to the group Aut(B,A) := {ϕ ∈ Aut(B) : ϕ(A) = A}. In particular, the group Aut(B,A) is isomorphic to Aut(N) if the group Aut(N) is abelian. The next result from [15] shows the total number of PGs which admit HGS of cyclic case of degree pqw. Theorem 3. The total number of PGs G which admits HGS of cyclic case is 12(r + i+ 1)[σ0(s) + σ1(j) + σ0(s)σ1(j)] of isomorphism types of degree pqw. The regular group is the cyclic of order pqw. Any field extension L/K admits a unique HGS of cyclic case G and is essentially classically Galois and for all groups G. More general, since any Sylow subgroup is cyclic. Hence, the square free groups of order n can exist and classify. B. Jamal, A. Alabdali / Eur. J. Pure Appl. Math, 16 (2) (2023), 1118-1127 1122 3. Results We will concentrate the rest of the work on HGSs on SFEs of degree pqw with p = 2w + 1, q and w are odd primes of square free. As a result, w and p are Sophie Germain prime and safe prime, respectively. We have w− 1 = 2ij, q− 1 = 2rs with i, r ≥ 1 and s, j are odd numbers. We write gcd(j, 2pw) = 1 and gcd(s, 2pq) = 1. However, we have no more presumptions regarding the prime factors of j and s. There are six groups N of order pqw up to isomorphism, but in this work we deal with the CG Cpqw and precisely the nonabelian case. As a result, the transitive subgroups of Hol(Cpqw) must be determined. Assume that N is a CG of order pqw with the following form N = ⟨σ, τ : σe = τw = 1, τσ = στ⟩, where e = pq. We write Aut(N) ∼= Aut(⟨σ⟩)× Aut(⟨τ⟩), since we have the two characteristic subgroups ⟨σ⟩ and ⟨τ⟩ in N , where Aut(⟨σ⟩) of order (p−1)(q−1) = 2w2rs and Aut(⟨τ⟩) of order w− 1 = 2ij are cyclic. Suppose that α, β, γ, δ are automorphisms of the group N of order w, 2, 2r, s respectively that make τ fix, and assume that η, θ are automorphisms of the group N of order 2i, j respectively that fix σ. The direct product ⟨α⟩⟨β, γ, η⟩⟨δ, θ⟩ is decomposed by Aut(N), where the factors have coprime orders w, 2(r+i+1) and sj respec- tively. A subgroup of Aut(N) decomposes into one subgroup from each of these factors as a direct product. The number of divisors in s is σ0(s), σ1(j) in j and σ0(s)σ1(j) in sj. Proposition 3. Let Jl = ⟨σ, [τ, αl]⟩ with 1 ≤ l ≤ w − 1. Then Jl is a nonabelian reg- ular subgroup of Hol(N). In addition, Table 1 shows the transitive subgroups G for the nonabelian group Jl of Hol(N). Proof. It is clear that Jl is nonabelian and regular of order pqw on N , since we discover that [τ, αl] has order w (since α fixes τ) and [τ, αl]σ = αl(σ)[τ, αl] in Jl. Given that holomorph of the N has a subgroup Z = ⟨σ, τ, α⟩ of order pqw2 with index 2ij relatively prime to pqw uniquely. So pqw divides the order of any transitive subgroup B, hence B ∩ Z must be transitive on N . As a result, either B ∩ Z is regular on N or B ⊂ Z. Now, there is one nonregular subgroup ⟨σ, α⟩ in Z and the other subgroups of order pqw are N and Jl. For some l, we have B ∩ Z = Z or N or Jl. That means each individual transitive subgroup B has either N or some Jl. So any subgroup of Aut(N) can be used to create a transitive subgroup B, since in Hol(N) the group N is normal . If ψ ∈ Aut(N) and ψ(τ) ̸= τ , we obtain ψ[τ, αl]ψ−1 = [ψ(τ), αl] /∈ Jl, so in Aut(N) the group ⟨α, β, γ2r−c1 , δs/d⟩ is the normalizer of Jl. As a result, any transitive subgroup B contains Jl, has the forms Jl, Jl ⋊ ⟨β⟩, Jl ⋊ ⟨γ2r−c1 ⟩ or Jl ⋊ ⟨δs/d⟩. Therefore, the Table 1 of transitive subgroups is derived from Aut(N) subgroups. Lemma 1. Table 1 shows that there are w − 1 groups in case 1 and 2 which are PGs and isomorphic. But, in cases 3 and 4 there are (w − 1)(r + 1) and (w − 1)σ0(s) groups respectively which are isomorphic as PGs. B. Jamal, A. Alabdali / Eur. J. Pure Appl. Math, 16 (2) (2023), 1118-1127 1123 Proof. Let 1 ≤ l ≤ w − 1 and ψ ∈ Aut(N) with ψ(τ) = τ l be the two variables. Consequently, ψ[τ, αl]ψ−1 = [ψ(τ), αl] = [τ, α]l. In addition, ψβψ−1 = β. As a result, conjugating by ψ yields the isomorphism Jl ⋊ ⟨β⟩ → J1 ⋊ β, which is a PG isomorphism because the stabilizer ⟨β⟩ of 1N is fixed. It also determines to Jl → J1 isomorphism. As a result, in case (2) all the groups are PGs as are isomorphic, and simultaneously (1). Let 1 ≤ l ≤ w − 1, 0 ≤ c1 ≤ r so ψγ2 r−c1ψ−1 = γ2 r−c1 and then conjugating by ψ obtains the isomorphism Jl⋊⟨γ2r−c1 ⟩ → J1⋊γ2 r−c1 . Finally, let 1 ≤ l ≤ w−1, d | s so ψδs/dψ−1 = δs/d and then conjugating by ψ obtains the isomorphism Jl ⋊ ⟨δs/d⟩ → J1 ⋊ δs/d. As a result, all the groups in case (3) and case (4) are PGs as are isomorphic. Table 1: The transitive subgroups for the nonabelian group Jl. Key Order Parameters # Groups Groups 1 pqw 1 ≤ l ≤ w − 1 w − 1 Jl 2 2pqw 1 ≤ l ≤ w − 1 w − 1 Jl ⋊ ⟨β⟩ 3 2c1pqw 1 ≤ l ≤ w − 1, 0 ≤ c1 ≤ r (w − 1)(r + 1) Jl ⋊ ⟨γ2r−c1 ⟩ 4 pqwd 1 ≤ l ≤ w − 1, d | s (w − 1)σ0(s) Jl ⋊ ⟨δs/d⟩ Lemma 2. The number of HGSs for the nonabelian group Jl is as in Table 3. Proof. In the cases are shown in Table 3, the stabilizer of 1N in B is B ′ = B∩Aut(N). We start with case 1, a single isomorphism class is formed by the w − 1 regular groups Jl. The automorphism ψ of Jl must induce an automorphism of the characteristic sub- group Cpqw in order for ψ to be compatible with the relation τσ = σuτ, u > 1, therefore ψ(σ) = σa and ψ(τ) = σbτ c with 1 ≤ a ≤ (q − 1)(p− 1), 0 ≤ b ≤ (q − 1)(p− 1), and 1 ≤ c ≤ (w − 1) are required c = 1. Thus | Aut(Jl) |= (p − 1)(q − 1)[(p − 1)(q − 1) + 1], and we have | B′ |= 1 for B = Jl. As a result, the number of cyclic HGSs on a Jl-extension by using Byott’s formula (1) is (p− 1)(q − 1) + 1. We assume in case 2 that B = Jl ⋊ ⟨β⟩ with Jl instead of N and B ′ = ⟨β⟩, we use Propo- sition 2. Conjugating by β fixes the generator F = [τ, αl] of order w by inverting σ. If ψ ∈ Aut(Jl), we have ψ(σ) = σa and ψ(F ) = σbF for 1 ≤ a ≤ (q − 1)(p− 1) ≤ and 0 ≤ b ≤ (q − 1)(p− 1) respectively. Then, b = 0 if and only if ψ normalizes B ′ in Aut(Jl). As a result, | Aut(B,B′ ) |= (p− 1)(q − 1), and the w − 1 conjugate subgroups yield that the number of HGS is 1. Table 2: The structures of transitive subgroups for Jl. Key Restrictions Order Structure 1 pqw Cpq ⋊ Cw 2 2pqw Cpq ⋊ C2w 3 c1 ̸= (0, 1), c1 = 0, c1 = 1 2c1pqw, pqw, 2pqw Cpq ⋊ C2c1w, Cpq ⋊ Cw, Cpq ⋊ C2w 4 d ̸= 1, d = 1 pqwd, pqw Cpq ⋊ Cwd, Cpq ⋊ Cw B. Jamal, A. Alabdali / Eur. J. Pure Appl. Math, 16 (2) (2023), 1118-1127 1124 Table 3: The number of HGSs for the nonabelian group Jl. Key Order | Aut(B,B′ ) | # iso. class # HGS per iso. class 1 pqw (p− 1)(q − 1)[(p− 1)(q − 1) + 1] 1 (p− 1)(q − 1) + 1 2 2pqw (p− 1)(q − 1) 1 1 3 2c1pqw (p− 1)(q − 1) r + 1 r + 1 4 pqwd (p− 1)(q − 1) σ0(s) σ0(s) We assume in cases 3 and 4 that B = Jl⋊⟨γ2r−c1 ⟩ and B = Jl⋊⟨δs/d⟩ respectively with Jl instead of N,B ′ = ⟨γ2r−c1 ⟩ and B′ = ⟨δs/d⟩ respectively, we also use Proposition 2. Con- jugating by ⟨γ2r−c1 ⟩ in case 3 and ⟨δs/d⟩ in case 4 fixes the generator F = [τ, αl] of order w by reversing σ. If ψ ∈ Aut(Jl), we have ψ(σ) = σa and ψ(F ) = σbF for 1 ≤ a ≤ (q−1)(p−1) and 0 ≤ b ≤ (q−1)(p−1) respectively. Then, b = 0 if and only if ψ normalize B ′ in Aut(Jl). As a result, in case 3 and case 4 | Aut(B,B′ ) |= (p− 1)(q − 1). (w − 1)(r + 1) in case 3 and (w − 1)σ0(s) in case 4 conjugate subgroups yield that the number of HGS in case 3 is (r + 1) and in case 4 is σ0(s). The conclusions of the nonabelian group are summed up in the theorem below. Theorem 4. The total number of isomorphism types admits nonabelian HGSs is 2+ (r+ 1) + σ0(s) of PGs of degree pqw. The nonabelain of order pqw is the regular group. Proof. It is clear from summing the numbers of permutation groups G of degree pqw of isomorphism types in column four from Table 3 that the total number is 2+(r+1)+σ0(s) which admits HGS of nonabelain case. The following theorem summarizes the results of the cyclic case in Theorem 3 and the nonabelain case in Theorem 4. Theorem 5. A HGS of cyclic type can realise isomorphism types in total 12(r + i + 1)[σ0(s) + σ1(j) + σ0(s)σ1(j)] + 2 + (r + 1) + σ0(s) of PGs G of degree pqw of both cases regular groups (where the Galois extensions have 1 HGS for the cyclic group and (p − 1)(q − 1) + 1, 1, r + 1, and σ0(s) HGSs for the nonabelian group of the cyclic type). Proof. It is clear from summing the numbers of PGs G of degree pqw of isomorphism types in column four from Table 3 and Theorem 3 that the total number is 12(r + i + 1)[σ0(s)+σ1(j)+σ0(s)σ1(j)]+2+(r+1)+σ0(s) which admits HGS of cyclic and nonabelain type G. Example 1. Assume that we have q = 5, w = 3, p = 2w + 1 = 7 three squarefree prime numbers. So, we have the conditions and notations of the group N as follow according to the primes above. q − 1 = 2r.s, r ≥ 1, s odd =⇒ q − 1 = 5 − 1 = 4 = 22.1 =⇒ r = 2, s = 1. Then d | s has d = 1 =⇒ σ0(s) = 1. w − 1 = 2i.j, i ≥ 1, j odd =⇒ w − 1 = 3− 1 = 2 = 21.1 =⇒ i = 1, j = 1. 0 ≤ c1 ≤ r =⇒ 0 ≤ c1 ≤ 2 that means c1 = 0, 1, 2. B. Jamal, A. Alabdali / Eur. J. Pure Appl. Math, 16 (2) (2023), 1118-1127 1125 1 ≤ l ≤ w − 1 =⇒ 1 ≤ l ≤ 2 that means l = 1, 2. As a result of these conditions, Table 1 and Table 3 have the following shape. Table 4: The transitive subgroups for the nonabelian group Jl when p = 7, q = 5, w = 3. Key Order Parameters # Groups Groups 1 105 l = 1, 2 2 J1, J2 2 210 l = 1, 2 2 J1 ⋊ ⟨β⟩, J2 ⋊ ⟨β⟩ 3 105 l = 1, 2, c1 = 0, 1, 2 6 J1 ⋊ ⟨γ4⟩, J2 ⋊ ⟨γ4⟩, 210 J1 ⋊ ⟨γ2⟩, J2 ⋊ ⟨γ2⟩, 420 J1 ⋊ ⟨γ⟩, J2 ⋊ ⟨γ⟩ 4 105 l = 1, 2, d | s = 1 | 1 2 J1 ⋊ ⟨δ⟩, J2 ⋊ ⟨δ⟩ Table 5: The number of HGSs for the nonabelian group Jl when p = 7, q = 5, w = 3. Key Order | Aut(B,B′ ) | # iso. class # HGS per iso. class 1 105 600 1 25 2 210 24 1 1 3 105, 210, 420 24 3 3 4 105 24 1 1 We can see from Theorem 4 that there is 6 isomorphism types admits nonabelian HGSs of PGs of degree 105. According to the results obtained in Theorem 5, A HGS of cyclic type can realise 150 isomorphism types of PGs G of degree 105 of both cases regular groups (cyclic and nonabelian), where the extension has 1 HGS for the cyclic groups and 25, 1, 3, 1 HGSs for the nonabelian groups of the cyclic type. 4. Discussion Comparing the work to the results of other references, we can see from the tables and results that similar behaviour exists for square free degree n = pqw as the field extension of degree n = pq in [6]. It is clear through Tables 1, 2 and 3 that no similar abstract group can be found for any two distinct PGs admitted HGSs. Thus we partially answer the question in [6] related to the behaviour of square free degree in general. 5. Conclusion We investigate the group permutations G for the nonabelian case of degree pqw where q, w ≥ 3 and p = 2w + 1 are all square free primes then for each G we enumerate the HGSs. There exists four G such that the field extensions L/K admit the HGSs in this case. 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