#### Abstract

Let *G* be a finite group and be the class of all finite supersoluble groups. A supersoluble subgroup *U* of *G* is called -maximal in *G* if for any supersoluble subgroup *V* of *G* containing *U*, . Moreover, is the intersection of all -maximal subgroups of *G*. This paper obtains some new criteria on , by assuming that some subgroups of *G* are either -*I*-supplemented or -*I*-embedded in *G*. Here, a subgroup *H* of *G* is called -*I*-supplemented in *G* if there exists a subnormal subgroup *T* of *G* such that and and -*I*-embedded in *G* if there exists a *S*-quasinormal subgroup *T* of *G* such that is *S*-quasinormal in *G* and .

#### 1. Introduction

As we know, the model of chemical substances, such as crystal, is a graph, whose change process can be represented by symmetric groups or others. Therefore, group theory plays an important role in chemistry and physics ([1, 2]). However, this paper focuses on a question in group theory, which will promote its development and, consequently, contribute chemistry and physics in many ways.

Throughout this paper, all groups are finite. *G* always denotes a group, denotes a prime, and is the set of all prime divisors of . All unexplained notation and terminology are standard, as in [3, 4, 5].

Recall that a class of groups is called a formation if is closed under taking homomorphic images and subdirect products. A formation is said to be (1) saturated, if whenever and (2) hereditary, if whenever . Following ([3], Chap. III, Definition 3.1), a subgroup *U* of *G* is called -maximal in *G* provided that (1) and (2) if and , then . Moreover, [6] denotes the intersection of all -maximal subgroups of *G*. As we know, the -hypercentre of *G* is the largest normal subgroup of *G* such that each *G*-chief factor below satisfies . Clearly, for any group *G* (see ([6], Theorem C)).

Let be a hereditary saturated formation and *N* be a normal subgroup of *G* contained in . Then, the following holds (1) for any subgroup *A* of *G* with and (2) for any subgroup *T* of *G* with . It is well known that the extensive applications of -hypercentral subgroups are based on the above properties, and there are two main topics about -hypercentre: the influence of -hypercentral subgroups on the structure of finite groups; the criteria of -hypercentral subgroups.

However, in [6], Theorem C shows that the above two properties still hold when , instead of the stronger condition . Therefore, it would be rather natural and of great significance to study . In fact, some recent results in this topic can be found in, for example, [6, 7, 8–12]. Particularly, in [6, 10], the authors have shown that in general, given the condition under which holds for every group *G*.

In connection with the topic of , a question naturally arises as follows:

*Question 1. *Can we give a condition under which a normal subgroup of G is contained in ?

In [9], Chen et al. gave some conditions under which a normal subgroup of *G* contained in . In this paper, we still pay attention to Question 1. Furthermore, we explore new criteria by the help of the following notion.

*Definition 1. *Let *H* be a subgroup of a group *G*. Then, *H* is called(1)*-I*-supplemented in *G* if there exists a subnormal subgroup *T* of *G* such that and (2)*-I-*embedded in *G* if there exists a *S*-quasinormal subgroup *T* of *G* such that is *S*-quasinormal in *G* and Our main results are the following:

Theorem 1. *Let E be a normal subgroup of G. For every prime and every noncyclic Sylow -subgroup of E, assume that all maximal subgroups of are either -I-supplemented or -I-embedded in G. Then .*

Theorem 2. *Let E be a normal subgroup of G. For every prime and every noncyclic Sylow -subgroup of E, assume that all cyclic subgroups of with order and 4 (when is a nonabelian 2-group) are either -I-supplemented or -I-embedded in G. Then, .*

#### 2. Preliminaries

Lemma 1 (see [13], Chapter 1 or [4], Chapter 1, Lemmas 5.34 and 5.35]). *Assume that H is a subgroup of G, , and .*(1)

*If*(2)

*H*is*S*-quasinormal in*G*, then is*S*-quasinormal in*E**If*(3)

*H*is*S*-quasinormal in*G*, then is*S*-quasinormal in*Assume that*(4)

*H*is a -group, then*H*is*S*-quasinormal in*G*if and only if*The set of*(5)

*S*-quasinormal subgroups of*G*is a sublattice of the subnormal subgroup lattice of*G**If*(6)

*H*is*S*-quasinormal in*G*, then is nilpotent*If*

*H*is a*π*-group and*H*is subnormal in*G*, thenLemma 2 ([6], Theorem C). *Let be a nonempty hereditary saturated formation. Assume that H, E, and N are subgroups of G with .*(1)

*(2)*

*(3)*

*If , then*(4)

*If , then*(5)

Lemma 3. *Assume that H is a -I-supplemented (resp., -I-embedded) subgroup of G.*(1)

*If*(2)

*N*is a normal subgroup of*G*satisfying either or , then is -*I*-supplemented (resp., -*I*-embedded) in*If*

*K*is a subgroup of*G*containing*H*, then*H*is -*I*-supplemented (resp., -*I*-embedded) in*K**Proof. *As the proof for -*I*-embedded subgroups is similar, we just assume that *H* is -*I*-supplemented in *G*. Then, *G* has a subnormal subgroup *T* such that and .(1)Clearly, is a subnormal subgroup of *G* such that . Consider . If , then by the modular law. Assume that . Then , which implies that . Thus, in both cases. Note that , and there exists the isomorphism: So from ([3], Chap. A, Theorem 9.2(e)) and Lemma 2(1), it follows that . By the definition, is -*I*-supplemented in .(2)Assume that . Clearly, is subnormal in *G* and . Note that and the isomorphismTherefore, by ([3], Chap. A, Theorem 9.2(e)) and Lemma 2(2), . So *H* is -*I*-supplemented in *K*.

Lemma 4 ([9], Lemmas 2.3 and 2.8). (1)*Let be a prime divisor of with . Then, , where denotes the class of all -nilpotent groups.*(2)*Assume that is a nonempty hereditary saturated formation. Let , be normal subgroups of G, and . Then, .*

Lemma 5. (1)*If T is a subnormal subgroup of G such that is a power of , then ([13], Lemma 1.1.11).*(2)

*Assume that*

*N*is a minimal normal subgroup of*G*. Then, or .*Proof. *If , then . Note that is a minimal normal subgroup of the -group . So the *G*-isomorphism shows that .

Lemma 6 ([3], Chap. A, Lemma 8.4). *Let N be a nilpotent normal subgroup of G and M a maximal subgroup of G satisfying . Then, is a normal subgroup of G.*

Lemma 7 ([5], Chap. VI, Theorem 4.7). *Let be a Sylow -subgroup of G and N a normal subgroup of G. If , then N is -nilpotent.*

#### 3. Proofs of Main Theorems

The following two propositions are main steps in the proof of Theorems 1 and 2, which also have independent meanings (see Corollaries 1 and 2).

Proposition 1. *Let be a Sylow -subgroup of G, where is a prime divisor of with . Assume that all maximal subgroups of are either -I-supplemented or -I-embedded in G. Then, G is -nilpotent.*

*Proof. *Suppose that the assertion is false and let *G* be a counterexample for which is minimal. We proceed via the following steps:(1)*G* has the unique minimal normal subgroup.* *Let *N* be a minimal normal subgroup of *G*. Assume that is an arbitrary maximal subgroup of , which is a Sylow -subgroup of . Then . Denote . Since , is a maximal subgroup of . By the hypothesis, *G* has a subnormal (resp., *S*-quasinormal) subgroup *T* such that is -*I*-supplemented (resp., -*I*-embedded) in *G*. Note that is a Sylow -subgroup of *N*, so is a -number. However, is a -number. So we have and then . Similarly as Lemma 3, it is easy to show that is a subnormal (resp., *S*-quasinormal) subgroup of such that is -*I*-supplemented (resp., -*I*-embedded) in . Therefore, satisfies the hypothesis. So the choice of *G* implies that is -nilpotent. Consequently, *N* is the unique minimal normal subgroup of *G*.(2) and .* *If , then the uniqueness of *N* implies that . In the case, is -nilpotent and so is *G*, a contradiction. Keep Lemmas 2(3) and 5(1) in mind. It is easy to obtain that .(3). If , then and . Clearly, . Then, there exists a maximal subgroup *M* of *G* such that . Together with the uniqueness of *N*, . Note that by Lemma 6, so and, consequently, . Here, and then . Thus, has a maximal subgroup such that . Clearly, and by the hypothesis, is either -*I*-supplemented or -*I*-embedded in *G*. First assume that is -*I*-supplemented in *G*. Combining with (2), there exists a subnormal subgroup *T* of *G* such that and . According to Lemma 5, we have , which deduces that . In this case, , a contradiction. Now suppose that is -*I*-embedded in *G*, that is, there exists a *S*-quasinormal subgroup *T* of *G* such that is *S*-quasinormal in *G* and . If , then is *S*-quasinormal in *G*. From Lemma 1(3) and the choice of , we deduce that , a contradiction. So . We further assume that . By Lemma 1(5), is nilpotent. Combining with (2), is a -group. Hence , which implies that , that is, is a *S*-quasinormal -subgroup of *G*. From Lemma 1(4)(6), it follows that , which shows that and consequently . In this case, by Lemma 1(3), T is normal in *G*. Hence, the minimality of *N* implies that . Consequently, , which shows that , a contradiction. Therefore, and the uniqueness of *N* implies . Consequently, . Similarly as the above, it is impossible. So we should assume that .(4)Final contradiction. Assume that , that is, *G* is a simple group. For any maximal subgroup of , if *T* is a subnormal (resp., *S*-quasinormal) subgroup of *G* such that is -*I*-supplemented (resp., -*I*-embedded) in *G*, then . As a result, , that is, , a contradiction. Therefore, . If , then *N* satisfies the hypothesis by Lemma 3(2). So the choice of *G* and the relationship deduce that *N* is -nilpotent, which contradicts (2) and (3). In general, we conclude . Let be a maximal subgroup of containing . Then and is either -*I*-supplemented or -*I*-embedded in *G*. First assume that is -*I*-supplemented in *G*. So there exists a subnormal subgroup *T* of *G* such that and . According to Lemma 6, we have that . Hence, . Note that and . So . However, it deduces that *N* is -nilpotent by Lemma 7, a contradiction. If is -*I*-embedded in *G*, then *G* has a *S*-quasinormal subgroup *T* such that is *S*-quasinormal in *G* and . Note that implies that is *S*-quasinormal in *G*, which contradicts (3) and Lemma 1(4)(6). Moreover, if , then by Lemma 1(5) and the uniqueness of *N*, and *N* is nilpotent, which contradicts (1) and (3). Therefore, and consequently . In this case, we finally conclude that . By Lemma 7, we also have that *N* is -nilpotent, a contradiction. This contradiction completes the proof.

Proposition 2. *Let be a Sylow -subgroup of G, where is a prime divisor of with . Assume that all cyclic subgroups of of order and order 4 when is a nonabelian 2-group are either -I-supplemented or -I-embedded in G. Then, G is -nilpotent.*

*Proof. *Suppose that the assertion is false and let *G* be a counterexample of minimal order.

Let *M* be a proper subgroup of *G* and a Sylow -subgroup of *M*. Then, for some . Now consider , which has a Sylow -subgroup . By Lemma 3(2), satisfies the hypothesis for *G*. So is -nilpotent by the minimality of *G*. Consequently, *M* is -nilpotent, and *G* is a minimal non--nilpotent group. By [14], Theorem 3.4.11, the following hold (i) , where and *Q* is a Sylow *q*-subgroup of *G* with ; (ii) is a noncyclic *G*-chief factor; (iii) the exponent of is or 4 (when is a nonabelian 2-subgroup). Take , and denote . Then, *H* has order or 4, , and . Moreover, *H* is either -*I*-supplemented or -*I*-embedded in *G* by the hypothesis.

First assume that *H* is -*I*-embedded in *G*. Let *T* be a *S*-quasinormal subgroup of *G* such that is *S*-quasinormal in *G* and . Note that is a *G*-chief factor, so we separate the proof into three cases: (1) ; (2); (3) and . If the first case holds, then and consequently . From Lemmas 2(1) and 4(1), we further deduce that , where denotes the class of all -nilpotent groups. So by (ii). Together with Lemma 2(3), we finally have that is -nilpotent, and so is *G*, a contradiction. In case (2), we have . Then is *S*-quasinormal in *G* by Lemma 1(2)(4). Note that is abelian. So according to Lemma 1(3). Consequently, by (ii), a contradiction. Lastly, suppose that case (3) holds, that is, and . If , then is p-nilpotent by Lemma 1(5). By (i), we have that is -nilpotent, and furthermore, is -nilpotent, which deduces that *G* is -nilpotent, a contradiction. So we have and is -nilpotent as *G* is a minimal non--nilpotent group. Note that , so . By Lemma 1(2), we obtain that is a *S*-quasinormal subgroup of contained in . Consequently, we deduce that from Lemma 1(3). Therefore, or , that is, or . However, from the proof of cases (1) and (2), we know that it is impossible.

Then assume that *H* is -*I*-supplemented in *G*, and *T* is a subnormal subgroup of *G* such that and . Clearly, by Lemma 5(1). So we have . Similarly as the first case above, we know that it is impossible. Thus, the assertion holds.

Now we true to prove Theorems 1 and 2.

*Proof of Theorem 1. *Suppose that the result is false and let be a counterexample for which is minimal. We proceed via the following steps.(1)*E* is a -group. Assume that , is the smallest prime divisor of and is a Sylow -subgroup of *E*. If is cyclic, then *E* is -nilpotent (see [15], Theorem 10.1.9). Now assume that is noncyclic. From the hypothesis and Lemma 3(2), it follows that *E* satisfies the hypothesis of Proposition 1. So we have that *E* is still -nilpotent. Let be the normal -Hall subgroup of *E*. Then is a normal subgroup of *G* and satisfies the hypothesis. Hence by the choice of . Suppose that is cyclic. Then, for the *G*-isomorphism . By Lemma 2(5), . Now assume that is noncyclic. By Lemma 3(1), we can easily obtain that satisfies the hypothesis. Analogously, the choice of implies that . Therefore, in any case, . Furthermore, by Lemma 2(4), a contradiction. Thus, , that is, *E* is a -group.(2). Since , there exists a -maximal subgroup *X* of *G* such that . By Lemma 3(2), satisfies the hypothesis for . If , then the choice of implies that . Note that is supersoluble for the isomorphism . So by Lemma 2(3). Furthermore, the choice of *X* implies , that is, . This contradiction shows that and, consequently, is supersoluble as .(3)*G* has the unique Sylow -subgroup. Let *q* be the largest prime dividing and a Sylow *q*-subgroup of *G*. Assume that . Note that is supersoluble. So and . Consider , which satisfies the hypothesis by Lemma 3(2). Note that is the smallest prime divisor of and *E* is the Sylow -subgroup of . So by Proposition 1, is -nilpotent. Therefore, and, consequently, . Now consider , which satisfies the hypothesis by Lemma 3(1). So the choice of implies that . Moreover, the isomorphism deduces that is supersoluble. Together with Lemma 2(3), we finally obtain is supersoluble. Furthermore, *G* is supersoluble by the isomorphism . This contradiction shows . So (3) holds.(4)Final contradiction. Let *N* be a minimal normal subgroup of *G* contained in *E*. Consider , which satisfies the hypothesis by Lemma 3(1). So the choice of implies that . Note that is supersoluble by the isomorphism . Combining with Lemma 2(3), is supersoluble. Therefore, and *N* is the unique minimal normal subgroup of *G* contained in *E*. Note that is a normal subgroup of *G* contained in *E*. So the uniqueness of *N* implies that . Consequently, *E* is the direct product of the minimal normal subgroups of *G* contained in *E* (see [14], Chap. 1, Lemma 1.8.17). Furthermore, by the uniqueness of *N*. Note that is a nontrivial normal subgroup of *G*. So , that is, . Take be an arbitrary maximal subgroup of *E*. Clearly, , and by the hypothesis, is either -*I*-supplemented or -*I*-embedded in *G*. Assume that is -*I*-supplemented in *G*. Let *T* be a subnormal subgroup of *G* such that and . Then, and, consequently, by the minimality of *E*. In this case, , which implies by the minimality of *E* again, a contradiction. Now suppose that is -*I*-embedded in *G* and *T* is a *S*-quasinormal subgroup of *G* such that is *S*-quasinormal in *G* and . It is easy to show that the above holds if *T* is replaced by . So, without loss of generality, assume that . Since *T* is *S*-quasinormal in *G*, we have by Lemma 1(3) and the relationship . Therefore, or . If , then is *S*-quasinormal in *G* and, similarly as above, , which contradicts the minimality of *E*. But if , then , which also deduces a contradiction as above. So the proof is completed.

*Proof of Theorem 2. *Suppose that the result is false and let be a counterexample for which is minimal. Then, *G* is not supersolvable. Similarly as steps (1) and (2) in the proof of Theorem 1, assume that and *E* is a -group.

Let *M* be any proper subgroup of *G*. Consider , which satisfies the hypothesis for by Lemma 3(2). So the minimality of deduces that . Note that is supersolvable by the isomorphism . So *M* is supersolvable by Lemma 2(3). Consequently, *G* is a minimal nonsupersolvable group and from ([14], Theorem 3.4.2), we deduce that (i) is a -subgroup of *G*; (ii) is a noncyclic *G*-chief factor; (iii) the exponent of *E* is or 4 (when *E* is a non-abelian 2-group). Similarly, as step (3) of the proof of Theorem 1, the Sylow -subgroup of *G* is normal in *G*. Note that is a nontrivial normal subgroup of , so .

Take , and denote . Then, *H* has order or 4, , and . By the hypothesis, *H* is either -*I*-supplemented or -*I*-embedded in *G*.

Assume that *H* is -*I*-embedded in *G*. Let *T* be a *S*-quasinormal subgroup of *G* such that is *S*-quasinormal in *G* and . Clearly, is another *S*-quasinormal subgroup of *G* such that *H* is -*I*-embedded in *G*. So without loss of generality, assume that . Then, is a *S*-quasinormal subgroup of contained in . Together with Lemma 1(3) and the relationship , we have . Thus, or . If , then is *S*-quasinormal in *G* and then . The choice of *H* shows that , which contradicts (ii). Assume that . Then, and by Lemma 2(2), . Together with (ii), we have . Recall that is supersoluble. Therefore, is supersoluble by the isomorphism and Lemma 2(3). Furthermore, we have *G* is supersoluble, a contradiction.

Now assume that *H* is -*I*-supplemented in *G* and *T* is a subnormal subgroup of *G* such that and . It is easy to show that . So or . Similarly as the above, is impossible. However, implies that , which contradicts (ii). Then, we complete the proof.

#### 4. Some Applications

The following result follows directly from Lemma 2(3) and Theorems 1 and 2.

Corollary 1. *Let E be a normal subgroup of G such that is supersoluble. Then, G is supersoluble if and only if for every prime and every noncyclic Sylow -subgroup of E, one of the following holds *(1)

*(2)*

*All maximal subgroups of**are either*-*I*-*supplemented or*-*I*-*embedded in**G*

*All cyclic subgroups of**with order**and*4 (*when**is a non-abelian 2-group*)*are either*-*I*-*supplemented or*-*I*-*embedded in**G*Recall also that a subgroup *H* of *G* is called as follows: *c*-normal in *G* [16] if *G* has a normal subgroup *T* such that and ; -normal in *G* [17] if *G* has a subnormal subgroup *T* such that and ; -supplemented in *G* [18] if *G* has a subnormal subgroup *T* such that and . Obviously, *c*-normal subgroups, -normal subgroups, and -supplemented subgroups of *G* are all -*I*-supplemented in *G*. Hence, we have the following.

Corollary 2. * G is supersoluble, if one of the following holds*(a)

*Every maximal subgroup of every Sylow subgroup of*(b)

*G*is -normal in*G*([17], Corollary 1.3)*All cyclic subgroups of*(c)

*G*with prime order or order 4 are -normal in*G*([17], Corollary 1.5)*Every maximal subgroup of every Sylow subgroup of*(d)

*G*is*c*-normal in*G*([16], Theorem 4.1)*All cyclic subgroups of*

*G*with prime order or order*4*are*c*-normal in*G*([16], Theorem 4.2)From Proposition 2, we obtain the following.

Corollary 3 ([19], Lemma 3.1). *Let be the smallest prime dividing and a Sylow -subgroup of G. If all subgroups of with order or order 4 are c-normal in G, then G is -nilpotent.*

Corollary 4 ([18], Theorem 3.1). *Let be a Sylow -subgroup of G, where is a prime dividing such that . If every maximal subgroup of is -supplemented in G, then G is p-nilpotent.*

Moreover, Theorem 3 in [20] and Theorems 3.3 and 3.4 in [21] follow directly from Theorem 1.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare no conflicts of interest regarding the content and implications of this manuscript.

#### Acknowledgments

This work was supported by the Start-up Scientific Research Foundation of Anhui Jianzhu University (2017QD20) and the Key Research Projects of Natural Science in Anhui Province (KJ2019A0784).