**INTRODUCTION**

All groups considered in this study will be finite and G always means a finite group. The conventional notions and notations, as in Doerk and Hawkes^{1}.

Recall that a formation is a hypomorph 𝔍 of groups such that each group G has the smallest normal subgroup whose quotient is still in 𝔍. A formation 𝔍 is said to be saturated if it contains each group G with G/Φ(G)∈𝔍. In this study, the symbol U denote the class of supersolvable groups. Clearly, U is a saturated formation. A formation 𝔍 is said to be S-closed (S_{n}-closed) if it contains every subgroup (every normal subgroup, respectively) of all its groups. Let [A]B stand for the semi-product of two groups A and B. For a class 𝔍 of groups, a chief factor H/K of a group G is called 𝔍-central^{2} if [H/K](G/C_{G} (H/K))∈𝔍. The symbol Z_{𝔍}(G) denotes the 𝔍-heypercenter of a group G, that is the product of all such H of G whose G-chief factors are 𝔍-central.

Recall that two subgroups H and K of a group G are said to permute if HK = KH. A subgroup H of a group G is called quasinormal (or permutable) in G if it permutes with all subgroups of G. A subgroup H of a group G is said to be c-normal in G^{3} if G has a normal subgroup N such that G = HN and H∩N__<__H_{G}, where H_{G} = Core_{G} (H) = ∩H_{g} is the maximal normal subgroup of G which is contained in H. Guo *et al*.^{4} introduced the following concept. They defined that the subgroup H of a group G is said to be 𝔍_{h}-normal if there exists a normal subgroup K of G such that HK is a normal Hall subgroup of G and (H∩K)H_{G}/H_{G}__<__Z_{𝔍} (G/H_{G}), the researchers have obtained some interesting results^{5}. In spite of the fact that the c-normal and 𝔍_{h}-normal are quite different generalizations of normality there are several analogous results which were obtained independently for c-normal and 𝔍_{h}-normal subgroups. Recently, Mohamed *et al*.^{6}, introduced the following concept which covers normality, c-normality and 𝔍_{h}-normality.

**Definition: **A subgroup H of G is 𝔍_{hq}-supplemented in G if G has a quasinormal subgroup N such that HN is a Hall subgroup of G and (H∩K)H_{G}/H_{G}__<__Z_{𝔍} (G/HG_{G}).

Several studies investigated the relationship between the properties of subgroups of a finite group G and the structure of G^{7-10}. Specially, maximal subgroups of sylow subgroups play an important role in determining the structure of a finite group. They have been studied by many scholars. A typical result in this direction is due to Srinivasan^{11}. It states that a group G is supersolvable if it has a normal subgroup N with supersolvable quotient G/N such that the maximal subgroups of the sylow subgroups of N are normal in G.

The main goal of this study is to report the structure of G under assumption that the maximal subgroups of the sylow subgroups of G are U_{hq}-supplemented in G and to discuss some applications.

**Preliminaries**

Lemma 2.1: Let G be a group and H__<__K__<__G. Then:

(a) |
H is 𝔍_{hq}-supplemented in G if and only if G has a quasinormal subgroup N such that HN is a Hall subgroup of G, H_{G}__<__N and (H/H_{G})∩(N/H_{G})__<__Z_{𝔍} (G/H_{G}) |

(b) |
If H is a normal subgroup of G and K is 𝔍_{hq}-supplemented in G, then K/H is 𝔍_{hq}-supplemented in G/H |

(c) |
If H is a normal subgroup of G, then the subgroup EH/H is 𝔍_{hq}-supplemented in G/H for every 𝔍_{hq}-supplemented in G subgroup E satisfying (∣H∣, ∣E∣) = 1 |

(d) |
If H is 𝔍_{hq}-supplemented in G and 𝔍 is S-closed, then H is 𝔍_{hq}-supplemented in K |

**Proof: **Guo^{2}

**Lemma 2.2: **If p_{n} is the smallest prime dividing the order of a group G and p_{1} is the largest prime dividing the order of G, where p_{n}≠p_{1}, then G possesses supersolvable subgroups H and K with ∣G:H∣ = p_{n} and ∣G:K∣ = p_{1} if and only if G is supersolvable.

**Proof: **Ramadan *et al*.^{14}

**RESULTS**

**Lemma 3.1:** Let p be the smallest prime dividing the order of G and let G_{p} be a sylow p-subgroup of G. If the maximal subgroups of G_{p} are U_{hq}-supplemented in G, then G is p-nilpotent.

**Proof:** Suppose the result is false and let G be a counter-example of minimal order. For the sake of clarity, the proof breaks into four parts:

Suppose O_{P}(G)≠1. Now consider the group G/O_{p}(G). Clearly G_{p}O_{p}(G)/O_{p}(G) is a sylow p-subgroup of G/O_{p}(G). Let PO_{p}(G)/O_{p}(G) be a maximal subgroup of G_{p}O_{p}(G)/O_{p}(G). Then P is a maximal subgroup of G_{p}. By hypothesis, P is U_{hq}-supplemented in G. So PO_{p}(G)/O_{p}(G)is U_{hq}-supplemented in G/O_{p}(G), by Lemma 2.1, then the hypothesis of theorem hold on G/O_{p}(G). Hence, G/O_{p}(G) is p-nilpotent by the minimality of G and so does G; a contradiction.

Suppose Z_{U} (G)≠1. If Z_{U} (G) is not p-subgroup of G, then Z_{U} (G) has a normal sylow q-subgroup Q such that q is the largest prime dividing the order of Z_{U} (G), as Z_{U} (G) is supersolvable. Clearly q≠p. Since Q characteristic in Z_{U} (G) and Z_{U} (G) is a normal subgroup of G, it follows that Q is a normal subgroup of G. Then 1≠Q__<__O_{p}(G), a contradiction with 1. Now, it follows that Z_{U} (G) is a p-subgroup of G, hence there exists a normal subgroup N of G contained in Z_{U} (G) such that ∣N∣ = p. Consider the group G/N. Clearly G_{p}/N be a sylow p-subgroup of G/N. By hypothesis and Lemma 2.1 (b), the maximal subgroups of G_{p}/N are U_{hq}-supplemented in G/N. Now, it follows that G/N is p-nilpotent by the minimality of G, then G/N contains a normal p’-Hall subgroup K/N and since N is a cyclic subgroup of order p, it follows by Huppert^{13}, that K is p-nilpotent and also does G; a contradiction.

Suppose O_{p}(G) = 1. Then H_{G} = 1, for all subgroups H of G_{p}. Let P be a maximal subgroup of G_{p}. By hypothesis, P is U_{hq}-supplemented in G, then by Lemma 2.1 (a), there exists a quasinormal subgroup N of G such that PN is a Hall subgroup of G, P_{G}__<__N and P/P_{G}∩N/P_{G}__<__Z_{U} (G/P_{G}). Since P_{G} = 1, it follows that P∩N__<__Z_{U} (G) and since Z_{U}(G) = 1 by 2, it follows that P∩N = 1. Since PN is a Hall subgroup of G, it follows that P__<__G_{p}__<__PN and so G_{p} = P(G_{p}∩N). Now, it follows that ∣G_{p}∩N∣ = ∣G_{p}:P∣ = p and so G_{p}∩N is a cyclic sylow p-subgroup of N, then N is p-nilpotent by Huppert^{13}. Thus, there exists a normal p’-Hall subgroup H of N. Since N is quasinormal subgroup of G, it follows that N is subnormal subgroup of G. So, H is also subnormal subgroup of G. Since PN is a Hall subgroup of G and H is a p’-Hall subgroup of N, it follows that H is a p’-Hall subgroup of G, i.e., H is a subnormal p’-Hall subgroup of G. Now, it follows that H is a normal p’-Hall subgroup of G, then H = 1, as O_{P}(G) = 1 from 1. Thus N = G_{p}∩N is quasinormal subgroup of G of order p and so 1≠N__<__Z_{U}(G); a contradiction with 2.

**Final contradiction: **Let H be a minimal normal subgroup of G contained in O_{P}(G), then H≠1 as O_{P}(G)≠1 by 3. Clearly the hypothesis of the theorem can be hold on the group G/H, by Lemma 2.1 (b), then G/H is p-nilpotent by the minimality of G. Since the class of all p-nilpotent groups is a formation, it follows that H is a unique minimal normal subgroup of G contained in O_{P}(G). If H__<__Φ(G_{p})^{1}, H__<__Φ(G) and since G/H is p-nilpotent, it follows that G/Φ (G) is p-nilpotent. Now, it follows that G is also p-nilpotent as the class of all p-nilpotent groups is a saturated formation; a contradiction, then H is not subgroup of Φ(G_{p}). So, there exists a maximal subgroup P of G_{p} such that H is not subgroup of P. Clearly G_{p} = PH. By hypothesis P is U_{hq}-supplemented in G, then by Lemma 2.1 (a), there exists a quasinormal subgroup N of G such that PN is a Hall subgroup of G, P_{G}__<__N and P/P_{G}∩N/P_{G}__<__Z_{U}(G/P_{G}). Since H is a unique minimal normal subgroup of G contained in O_{P}(G), it follows that H__<__P_{G}__<__P; a contradiction. Thus P_{G} = 1, then it follows that P∩N<Z_{U}(G). But Z_{U} (G) = 1 by 2. Thus, P∩N = 1, then it follows that G_{P} = P(G_{P}∈N) and ∣G_{P}∩N∣ = ∣G_{P}:P∣ = p. By repeated the proof of 3, it follows that 1≠N__<__Z_{U}(G); a final contradiction with 2. As an immediate consequence of Theorem 3.1.

**Corollary 3.2: **If the maximal subgroups of the sylow subgroups of a group G are U_{hq}-supplemented in G except for the largest prime dividing the order of G, then G possesses an ordered sylow tower.

**Proof:** By Theorem 3.1, G is p-nilpotent, where p is the smallest prime dividing the order of G, then G = G_{p}K, where G_{p} is a sylow P-subgroup of G and K is a normal p’-Hall subgroup of G. By Lemma 2.1 (d), the hypothesis carries over K. Then K possesses an ordered sylow tower by the induction on the order of G, therefore; G possesses an ordered sylow tower.

Now prove that:

**Theorem 3.3: **Let P be a normal p-subgroup of a group G such that G/P∈U. If the maximal subgroups of P are U_{hq}-supplemented in G, then G∈U.

**Proof: **Suppose the result is false and let G be a counter-example of minimal order. Let G_{p} be a sylow p-subgroup of G. Treatment can be done by two cases:

**Case 1:** P = G_{p}

Then by Schur Zassenhaus’s Theorem, G/G_{p}≅K∈U, where K is a p’-Hall subgroup of G. Let N be a minimal normal subgroup of G contained in G_{p}. Then (G/N)/(G_{p}/N)≅G/G_{p}∈U. By hypothesis and Lemma 2.1 (b), the maximal subgroups of G/N are U_{hq}-supplemented in G/N. Then G/N∈U, by the minimality of G. Since the class of all supersolvable groups is a saturated formation, it follows that N is a unique minimal normal subgroup of G contained in G_{p}. If Φ (G_{p})≠1, then N<Φ(G_{p}) and so N<Φ(G) by Doerk and Hawkes^{1}. Clearly G/Φ(G)≅(G/N)/(Φ(G)/N)∈U. Then G∈U, as the class U is a saturated formation; a contradiction. Thus Φ(G_{p}). Now, it follows that there exists a maximal subgroup P_{1} of G_{p} such that N is not subgroup of P_{1}. By hypothesis P_{1} is U_{hq}-supplemented in G, then by Lemma 2.1 (a), there exists a quasinormal subgroup H of G, (P_{1})_{G}__<__h and (P_{1}/(P_{1})_{G})∩(H/(P_{1})_{G}__<__Z_{U}(G/(P_{1})_{G}). Since N is a unique minimal normal subgroup of G contained in G_{p}, it follows that N__<__(P_{1})_{G}__<__P_{1}; a contradiction. Thus (P_{1})_{G} = 1. Hence, P_{1}∩H__<__Z_{U} (G). If P_{1}∩H≠1, then G_{p}∩Z_{U}(G) be a non-trivial normal sylow p-subgroup of Z_{U}(G). Now, it follows that G_{p}∩Z_{U} is supersolvably embedded in G, then G_{P}∩Z_{U}(G) contains a subgroup L of order p is a normal subgroup of G. By the uniqueness and minimality of N, it follows that L = N. Then G∈U, as G/N∈U and ∣N∣ = p; a contradiction, thus P_{1}∩H = 1. Since, P_{1}H is a Hall subgroup of G, it follows that P_{1}__<__G_{p}__<__P_{1}H. Then G_{p} = P_{1}(G_{p}∩H) and ∣G_{p}∩H∣ = p as P_{1}∩H = 1. Also since H is quasinormal subgroup in G, it follows that HK__<__G, then G_{p}∩H = G_{p}∩HK is a normal sylow p-subgroup of HK. It follows that K__<__N_{G}(G_{p}∩H). Since Φ(G_{p}) = 1, it follows that G_{p} is an elementary abelian, then G_{p}__<__N_{G}(G_{p}∩H). Now, it follows that G = G_{p}K__<__N_{G}(G_{p}∩H), i.e., G_{p}∩H is a normal subgroup of G. By the uniqueness and minimality of N, it follows that G_{p}∩H = N and so ∣N∣ = ∣G_{p}∩H∣ = p. Then G∈U, as G/N∈U and ∣N∣ = p; a contradiction.

**Case 2: **P__<__G_{p}

Put π(G) = {p_{1}, p_{2},…, p_{n}} be a set of all primes dividing the order of G, where p_{1}>p_{2}>…>p_{n}. Since G/P∈U, it follows by Lemma 2.2, that G/P possesses two super solvable subgroups H/P and K/P with ∣G/P:H/P∣ = p_{1} and ∣G/P:K/P∣ = p_{n}. By Lemma 2.1 (d), the hypothesis carries over H/P and K/P. It follows that by the minimality of G, H and K are in U and ∣G:H∣ = ∣G/P:H/P∣ = p_{1} and ∣G:K∣ = ∣G/P:K/P∣ = p_{n}. Hence, by Lemma 2.2, G∈U; a final contradiction.

As corollaries of Corollary 3.2 and Theorem 3.3.

**Corollary 3.4: **Let K be a normal subgroup of a group G such that G/K∈U. If the maximal subgroups of the sylow subgroups of K are U_{hq}-supplemented in G, then G∈U.

**Proof: **By Lemma 2.1 (d), the maximal subgroups of the sylow subgroups of K are U_{hq}-supplemented in K. Then by Corollary 3.2, K possesses an ordered sylow tower. It follows that 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 of K and K is a normal subgroup of G, it follows that K_{p} is a normal subgroup of G. Now consider the factor group G/K_{p}. Since G/K∈U, it follows that (G/K_{p})/(K/K_{p})≅G/K∈U and since the maximal subgroups of the sylow subgroups of K are U_{hq}-supplemented in G, it follows by Lemma 2.1 (c), the maximal subgroups of the sylow subgroups of K/K_{p} are U_{hq}-supplemented in G/K_{p}. Then G/K_{p}∈U, by the induction on the order of G. Therefore G∈U, by Theorem 3.3.

**Corollary 3.5:** If the maximal subgroups of the sylow subgroups of a group G are U_{hq}-supplemented in G, then G∈U.

**Proof: **By Corollary 3.2, G possesses an ordered sylow tower, then G has a normal sylow p-subgroup G_{p}, where p is the largest prime dividing the order of G. By Lemma 2.1 (c), our hypothesis carries over G/G_{p}. Then G/G_{p}∈U, by the induction on the order of G. Therefore G∈U, by Theorem 3.3.

**Some applications: **Finally, consider some applications of Theorems 3.1, 3.3 and Corollaries 3.4, 3.5.

**Corollary 4.1: **Let p be the smallest prime dividing the order of G and let G_{p} be a sylow p-subgroup of G. If the maximal subgroups of G_{p} are c-normal in G, then G is p-nilpotent^{14}.

**Corollary 4.2:** Let K be a normal subgroup of a group G such that G/K∈U. If the maximal subgroups of the sylow subgroups of K are normal in G, then G∈U^{11}.

**Corollary 4.3: **Let K be a normal subgroup of a group G such that G/K∈U. If the maximal subgroups of the sylow subgroups of K are c-normal in G, then G∈U^{14}.

**Corollary 4.4:** Let K be a normal subgroup of a group G such that G/K∈U. If the maximal subgroups of the sylow subgroups of K are U_{h}-normal in G, then G∈U^{14}.

**Corollary 4.5: **If the maximal subgroups of the sylow subgroups of a group G are normal in G, then G∈U^{11}.

**Corollary 4.6:** If the maximal subgroups of the sylow subgroups of a group G are c-normal in G, then G∈U^{3}.

**Corollary 4.7:** If the maximal subgroups of the sylow subgroups of a group G are U_{h}-normal in G, then G∈U^{4}.

**CONCLUSION**

This study improves and extends some results of super solvability of the group by using the concept of 𝔍_{hq}-supplemented.

**ACKNOWLEDGMENT**

The authors gratefully acknowledge the approval and the support of this study from the Deanship of Scientific Research study by the grant No. 8-19-1436-5 K. S. A., Northern Border University, Arar.