On non-normal cyclic subgroups of prime order or order 4 of finite groups

In this paper, we call a finite group G an NLM-group (NCM-group, respectively) if every nonnormal cyclic subgroup of prime order or order 4 (prime power order, respectively) in G is contained in a non-normal maximal subgroup of G. Using the property of NLM-groups and NCM-groups, we give a new necessary and sufficient condition for G to be a solvable T -group (normality is a transitive relation), some sufficient conditions for G to be supersolvable, and the classification of those groups whose all proper subgroups are NLM-groups.

. Let G be a group. We say that G has the NLM propertyshortly: G is an NLM-groupif each non-normal cyclic subgroup of prime order or order 4 in G is contained in a non-normal maximal subgroup of G.
The aim of this paper is threefold. First, we give a new characterization of solvable T -groups in terms of NCM property. We generalize Theorem 1 in [8].
Theorem A. A group G is a solvable T -group if and only if every subgroup of G is an NCM-group.
Next, we present a sufficient condition, which generalizes Theorem 7 in [8], for supersolvability of a group G.
Theorem B. Let G be a group such that all its non-nilpotent subgroups are NLM-groups. Then G is supersolvable.
Finally, we obtain a classification of those groups whose all proper subgroups are NLM-groups.
Theorem C. Let G be a non-nilpotent minimal non-NLM-group (non-NLM-group all of whose proper subgroups are NLM-groups). Then G must be one of the following types: (1) G is supersolvable.
(2) = ⋊ G P Q, where P is isomorphic to the quaternion group Q 8 , = ⟨ ⟩ Q z is cyclic of order > 3 1 r , z induces an automorphism in P such that / ( ) P P Φ is a faithful and irreducible Q-module, and z centralizes ( ) P Φ .
where i is a primitive q f th root of unity modulo p.

Preliminary results
In this section, we collect some necessary results used frequently in the proof of theorems. For the sake of clearness for proof, an easy lemma with no proof is as follows.
Lemma 2.1. Let = ⋊ G P Q, where P and Q are Sylow subgroups of G, P is normal in G and Q is abelian. Then all maximal subgroups of G are of type either PQ g 1 or P Q g 1 , where Q 1 is a maximal subgroup of Q and P 1 is such that / P P 1 is non-trivial and G-simple, where g varies in G and PQ g 1 is normal in G.
The main properties of minimal non-supersolvable groups appear in Doerk's paper [9].
Lemma 2.2. Let G be a minimal non-supersolvable group. Then (1) G has a unique normal Sylow p-subgroup P for some prime p.
If P is non-abelian and = p 2, then the exponent of P is 4. (6) If P is abelian, then the exponent of P is p.
where p is any prime, > m 0 and > n 0.
where P is the quaternion group of order 8, = ⟨ ⟩ Q y is cyclic of order > 3 1 r , y induces an automorphism permuting cyclically the three maximal subgroups of P.
, is an elementary abelian p-group of order p 2 , and = ⟨ ⟩ Q y is cyclic of order q r .

A necessary and sufficient condition for solvable T -groups
In this section, we will give the proof of Theorem A and its corollary.
Proof of Theorem A. It is obvious that a nilpotent group is an NCM-group if and only if it is a Dedekind group, i.e., its all subgroups are normal.
By [8, Theorem 1], we only need to prove the sufficient condition for the case that G is non-nilpotent. Assume that G is a counterexample of minimal order and every subgroup of G is an NCM-group. Surely, the assumption is inherited by subgroups. Consequently, every proper subgroup of G is a solvable T -group by the minimality of G.
Let G be of type (4) in Lemma 2.4. By Lemma 2.1, G has two kinds of maximal subgroups ( )⟨ ⟩ P y Φ g and Since y induces an automorphism permuting cyclically the three maximal subgroups of the quaternion group, three maximal subgroups of Q 8 are not normal in G. However, they are not contained in ( )⟨ ⟩ P y Φ g , a contradiction. Let G be of type (5) in Lemma 2.4. Then non-normal subgroup ⟨ ⟩ ab of G is only contained in ⟨ ⟩ P y q which is normal in G, a contradiction.
Let G be of type (6) in Lemma 2.4. In fact, G is minimal non-nilpotent and every subgroup of P is not normal in G. However, G has two kinds of maximal subgroups⟨ ⟩ y g and ⟨ ⟩ ⊴ P y G q , where ∈ g G. This induces a contradiction.
Let G be of type (7) in Lemma 2.4. Similar arguments as above, every subgroup of P is not normal in G and only contained in ⟨ ⟩ ⊴ P y G q , a contradiction. □ By [8, Theorem 1] and Theorem A, the following result is obvious.
Corollary 3.1. Let G be a group. Then the following conditions are equivalent: However, a group G in which all subgroups are NLM-groups is not necessarily a solvable T -group. 4 Sufficient conditions for a group G to be supersolvable We consider the structure of a group G by means of the NLM property. The following result plays a key role in proving later.
Lemma 4.1. Let = ⋊ G P Q be a supersolvable NLM-group, where P and Q are Sylow subgroups of G, P is elementary abelian and Q is cyclic non-normal in G. If all subgroups of G are NLM-groups, then G is a T -group.
Proof. Clearly, we only need to consider the case | | ≥ P p 2 for a prime p. Since G is supersolvable and P is elementary abelian, P can be regarded as a completely reducible Q-module over the field of p elements, . According to the structure of T -groups [11], if G is not a T -group, then there exist i, j such that ≠ α α i j . Take  Now we only need to consider the following two cases.
By Lemma 2.1, all maximal subgroups of G are either of the type PQ 1 or of the type ( ) P Q Φ g , where ∈ g G, and Q 1 is a maximal subgroup of Q. By the assumption that all non-nilpotent subgroups of G are NLM-groups and Lemma 2.2(3)(5)(6), the exponent of P is p or 4 if = p 2, and P has a cyclic subgroup N of order p or 4, which is not normal in G and ≰ ( ) N P Q Φ g . However, N is only contained in PQ 1 which is normal in G, a contradiction to the fact that G is an NLM-group as it is non-nilpotent. So G is supersolvable.
By the minimality of G, we have easily that p is the maximal prime divisor of ( ) π G and let > > p q r. Lemma 2.2(3) implies that the exponent of P is p.
(1) Assume that P is abelian.
Since the exponent of P is p, we have that P is an elementary abelian group and it is a minimal normal subgroup of G. If ⟨ ⟩ P x is nilpotent for any ∈ x Q, then Q is also normal in G, which contradicts the fact that minimal non-supersolvable has a unique normal Sylow subgroup. So there exists some element y such that ⟨ ⟩ P y is a non-nilpotent NLM-group, and ⟨ ⟩ P y is a T -group by Lemma 4.1. Thus, every subgroup of P is normal in PQ. Similarly, every subgroup of P is normal in PR, and is normal in G as well, a contradiction.
By Lemma 2.2(4), ( ) = ( ) = ′ P Z P P Φ . Note that the exponent of P is p, we have easily that / ( ) G P Φ is minimal non-supersolvable and it satisfies the condition of the theorem, and / ( ) is a minimal normal subgroup of / ( ) G P Φ . By induction, similar arguments to (1), / ( ) G P Φ is supersolvable, and so / ( ) G G Φ is supersolvable. Therefore, G is supersolvable, which is the final contradiction. □ The following two results are obvious by Theorem B.
Corollary 4.2. Let G be a group such that all its non-nilpotent subgroups are NCM-groups. Then G is supersolvable. 5 Non-NLM-groups whose proper subgroups are NLM-groups The final aim is to give the structure of minimal non-NLM-groups. Buckley [12] called a group G a PN -group if every minimal subgroup of G is normal in G and gave some characterizations of PN -groups. Sastry [13] investigated the structure of minimal non-PN -groups (non-PN -groups whose proper subgroups are PN -groups). Guo et al. [14] called a group G a * PN -group if all minimal subgroups and cyclic subgroups of order 4 of G are normal in G, and classified completely minimal non- * PN -groups (non- * PN -groups whose proper subgroups are * PN -groups). Obviously, a nilpotent group is an NLM-group if and only if it is a * PN -group. By combining [13,Theorem] and [14, Theorem 2.1], nilpotent minimal non-NLM-groups are as follows.
Theorem D. Let G be a p-group. Then G is a minimal non-NLM-group if and only if it is one of the following types: 1 , 1 , , p p p n n 1 , p an odd prime.
p n , p an odd prime.
n m n 1 , ≥ n 2, ≥ m 2, and at least one of n and m is 2.
A classification of non-nilpotent minimal non-NLM-groups is given in Theorem C, which will be proved as follows.
Proof of Theorem C. Suppose that G is a minimal non-NLM-group, then by Theorem B, every proper subgroup of G is supersolvable. Therefore, G is either supersolvable as type (1) or minimal non-supersolvable. In the following, we assume that G is minimal non-supersolvable.
By Lemma 2.2(1)(3)(5), G has a unique normal Sylow p-subgroup P, and the exponent of P is p or 4 if P is a non-abelian 2-group. Since all proper subgroups of G are NLM-groups, P is either elementary abelian or isomorphic to the quaternion group Q 8 .
First, we examine the classification of minimal non-supersolvable groups in [15,Theorem 9]. Groups of types 5, 7, 9, and 12 in [15, Theorem 9] all contain an extraspecial subgroup P of order p 3 , where > p 2. Note that P is not an NLM-group. Those groups are not satisfied with minimal non-NLM-groups. Let P be Q 8 as type 3 in [15,Theorem 9]. Note that / ( ) ≲ G C P S G 4 , where S 4 is the symmetric group of degree 4. It makes Q is a Sylow 3-subgroup of G. Furthermore, G is also a minimal non-nilpotent group, thus G is of type (2).
Next, we assume that G is isomorphic to one of types 2, 4 in [15,Theorem 9]. It is easy to see that all subgroups of G are ⟨ ⟩ P z q g and ⟨ ⟩ z g , where ∈ g G. In type 2, G is also a minimal non-nilpotent group, G is of type for ≤ ≤ − j q 0 1, z q induces a power automorphism with fixed-point-free on P. Certainly, all subgroups of ⟨ ⟩ P z q g are NLM-groups. Thus, G is of type (4). Then, we assume that G is isomorphic to one of types 6, 8, and 10 in [15,Theorem 9]. For arbitrary element x of E, by the assumption that ⟨ ⟩ P x is an NLM-group and Lemma 4.1, ⟨ ⟩ P x is a T -group. The arbitrariness of x leads to the normality of every subgroup of P in G, a contradiction.
Finally, let G be type 11 in [15,Theorem 9]. Namely, = G PQR, where ∈ ( ) P G Syl p , ⊴ P G, ∈ ( ) Q G Syl q , ∈ ( ) R G Syl r , R is a cyclic subgroup of order + r s t , with r a prime number and s and t integers such that ≥ s 1 and ≥ t 0, normalizing Q, / ( ) Q Q Φ is an irreducible R-module over the field of q elements, with kernel the subgroup D of order r t of R, and P is an irreducible QR-module over the field of p elements, where | − q p 1, | − r p 1 s , and | − r q 1. In this case, ( ) ′ G Φ p , the Hall ′ p -subgroup of ( ) G Φ , coincides with ( ) × Q D Φ and centralizes P. By the same arguments as above, PR is a T -group. Therefore, every subgroup of P is normal in PR. However, the supersolvability of PQ implies that there exists a non-trivial subgroup of P which is normal in PQ. This contradicts the fact P is an irreducible QR-module over the field of p elements. Therefore, G does not coincide with a minimal non-NLM-group. □