Finite groups with some NR-subgroups or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document}-subgroups

Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR-subgroup (Normal Restriction) if whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K \trianglelefteq H}$$\end{document}, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${K^G \cap H = K}$$\end{document}, where KG is the normal closure of K in G. Bianchi, Gillio Berta Mauri, Herzog and Verardi proved a characterization of soluble T-groups by means of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document}-subgroups: a subgroup H of G is said to be an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document}-subgroup of G if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^g \cap N_G(H) \leq H}$$\end{document} for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${g \in G}$$\end{document}. In this article we give new characterizations of finite soluble PST-groups in terms of NR-subgroups or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{H}}$$\end{document}-subgroups. We will show that they are different from the ones given by Ballester-Bolinches, Esteban-Romero and Pedraza-Aguilera. Robinson established the structure of minimal non-PST-groups. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PST-groups.


Introduction and notation
All groups considered in this paper are finite. We use conventional notions and notations, as in [4,10,18]. Throughout this article G stands for a finite group and π(G) denotes the set of primes dividing |G|. A subgroup H of G is said to be permutable in G if H permutes with every subgroup of G. A group G is said to be a PT -group (respectively, T -group) if permutability (respectively, normality) is a transitive relation in G. By a result of Ore [4] PT -groups are exactly those groups where all subnormal subgroups are permutable. A subgroup of G is called s-permutable in G if it permutes with all Sylow subgroups of G. A group G is said to be a PST-group if s-permutability is a transitive relation in G. By a result of Kegel ( [4], Theorem 1.2.14(3)) PST-groups are exactly those groups where all subnormal subgroups are s-permutable. In the literature there are several characterizations of finite soluble T -groups, PT -groups and PST-groups (see [3][4][5][6]8]).
A subgroup H of G is called a CR-subgroup (Character Restriction) of G if every complex irreducible character of H is a restriction of some irreducible character of G (see [12]). It is well known that if H is a CR-subgroup of G and K H , then K G ∩ H = K . In [7] Berkovich introduced an interesting subgroup embedding property: In [19] Tong-Viet showed that, if every maximal subgroup of G is an NR-subgroup of G, then G is soluble. In [7] Berkovich proved that, if all Sylow subgroups of a group G are NR-subgroups, then G is supersoluble.
For groups H T G we say that H is strongly closed in T with respect to G if H g ∩ T H for all g ∈ G. Bianchi, Gillio Berta Mauri, Herzog and Verardi ([8], Theorem 10) proved a characterization of soluble T -groups by means of H-subgroups: By [9] if H is a p-subgroup of G, then H is an H-subgroup of G if and only if H is strongly closed in P with respect to G for some Sylow p-subgroup P of G containing H . In [2] Asaad showed that if every maximal subgroup of every Sylow subgroup of G is an H-subgroup of G, then G is supersoluble.
The basic structure of soluble T -, PT -and PST-groups was established by Gaschütz, Zacher, and Agrawal; this result shows that the classes of all soluble T -, PT -, and PST-groups are closed under taking subgroups and under taking epimorphic images. Since a non-abelian simple group is a T -group the first assertion is not true for the class of all T -groups (PT -and PST-groups). For some other characterizations of these groups see [3][4][5][6]. The structure of minimal non-T -groups, minimal non-PTgroups and minimal non-PST-groups was established by Robinson [16,17]. By ( [6], Corollary 5 and Theorem 6) and [17] groups considered in [1,6,14] can be used for characterizations of minimal non-PST-groups. We give new characterizations of soluble T -, PT -, PST-groups and minimal non-PST-groups in terms of NR-subgroups or H-subgroups. We will show the differences between these characterizations and the ones given in [3][4][5][6].
Recall that a subgroup H of a group G is called a second maximal subgroup or a 2-maximal subgroup of G, if H is a maximal subgroup of some maximal subgroup of G. Second maximal subgroups were introduced by Huppert in [11] in which it was proved that a group is supersoluble if every its 2-maximal subgroup is normal. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PST-groups.

Preliminaries
Theorem 2.1 [4] Let L be the nilpotent residual of a group G. Then the following assertions hold.

(Agrawal) G is a soluble PST-group if and only if L is an abelian Hall sub-
group of odd order of G in which G acts by conjugation as a group of power automorphisms.

(Zacher) G is a soluble PT -group if and only if G is a soluble PST-group with
Iwasawa Sylow subgroups.

(Gaschütz) G is a soluble T -group if and only if G is a soluble PST-group with
Dedekind Sylow subgroups.

Lemma 2.2 Let p be a prime. Let N be a normal p -subgroup of a group G and P be any p-subgroup of G. Then P is an NR-subgroup of G if and only if P N /N is an NR-subgroup of G/N .
Proof Assume first that P is an NR-subgroup of G. Let M/N be a normal subgroup of P N/N . Then there exists a normal subgroup L of P such that M = L N . Since     (2, q), where q is odd, q > 3 and q ≡ 3 or 5 (mod 8); 3. SL (2, q), where q is odd, q > 3 and q ≡ 3 or 5 (mod 8).

Soluble T-, PT-and PST-groups
By Lemma 2.4 every NR-group is supersoluble. By Lemma 2.7 every H-group is supersoluble. The following example shows that there exists a supersoluble group which is neither an NR-group nor an H-group.
Example 3.1 Let p be an odd prime, P = a × b be an elementary abelian group of order p 2 . Let x be the automorphism of P of order 2 given by a x be the corresponding semidirect product. Then G is supersoluble. But G is neither an NR-group nor an H-group, since ab G ∩ P = P and By Lemma 2.6 every NR-group is an H-group. The following example shows that the converse does not hold.

Example 3.2 Let p be an odd prime and let
be an extraspecial group of order p 3 and exponent p. Let B = x be a cyclic group of order p and P = A × B. Let y be the automorphism of P of order 2 given by a y = a −1 , b y = b −1 , x y = x −1 . Let G = P y be the corresponding semidirect product. Then every maximal subgroup of P is normal in G, so is an The following example shows that the class of all NR-groups (the class of all H-groups) is not closed under taking subgroups.

Example 3.3 Let p be an odd prime and let
be an extraspecial group of order p 3 and exponent p. Let x be the automorphism of P order 2 given by a x be the corresponding semidirect product. It is easily seen that G is an NR-group and G is an H-group. Let Example 3.3 also shows that there exist NR-groups (H-groups) which are not soluble PST-groups.
Following [6,20] we say that G is a T 0 -group (respectively, a PT 0 -group, a PST 0group) if G/ (G) is a T -group (respectively, a PT -group, a PST-group). The following result was proved in [6]. Theorem 3.1 Let G be a group. The following statements are equivalent: The following examples show the differences between the classes of T 0 -, PT 0 -, PST 0 -groups and the classes of NR-, H-groups.
Example 3.4 Every non-abelian simple group is a T -group, a PT -group and a PSTgroup. Clearly by Lemmas 2.4 and 2.7 it is neither an NR-group nor an H-group, since it is not supersoluble.
Example 3.5 The group from Example 3.2 is a T 0 -group, a PT 0 -group, and PST 0group. It is supersoluble, but it is not an NR-group.
We do not know if there exists a soluble T 0 -group (a PT 0 -group, a PST 0 -group) that is not an H-group. Now we use NR-groups and H-groups to characterize soluble PST-groups. Theorem 3.2 Let G be a group. The following conditions are equivalent: 1. G is a soluble PST-group; 2. G and its subgroups are NR-groups; 3. G and its subgroups are H-groups.
Proof (1)⇒(2) We claim that every soluble PST-group is an NR-group. Assume that G is a soluble PST-group. Let L be the nilpotent residual of G. By Theorem 2.1 L is an abelian Hall subgroup of odd order of G in which G acts by conjugation as a group of power automorphisms. Let p ∈ π(L) and Since the class of all soluble PST-groups is closed under taking subgroups, it follows that G and its subgroups are NR-groups. (3)⇒(1) Assume that G is a group of minimal order satisfying that G and its subgroups are H-groups but G is not a PST-group. Then G is a minimal non-PST-group and so G = P Q, where P ∈ Syl p (G), Q ∈ Syl q (G), π(G) = {p, q} and Q is cyclic. We claim that every subgroup H of P is normalized by G q for any G q ∈ Syl q (G). Let H = H n H n−1 · · · H 0 = P be a series between H and P such that |H i : H i+1 | = p for i = 0, . . . , n − 1. Applying induction on n we may assume that n > 0. Since G q N G (H n−1 ) and so G q H n−1 is an H-group we have G q N G (H n ) by Lemma 2.5. Let G q be any Sylow q-subgroup of G and G q = x . Since G q N G (H ) for all H P, it follows that x induces the power automorphism on P. By ( [4], Lemma 1.3.4) P is abelian, since x induces a nontrivial automorphism on P and has order prime to p, so by ( [15], 13.4.3) the automorphism x is fixed-point-free. Hence P is not a 2-group. Therefore by Theorem 2.1 (1) G is a soluble PST-group, a contradiction. This ends the proof.
By Theorems 3.2 and 2.1 we obtain the following corollaries.

Theorem 4.2 Let G be a group of even order. Assume that every proper subgroup of even order of G is an NR-group. Then either G is an NR-group or |π(G)| 3.
Proof Assume that |π(G)| > 3. By Theorem 4.1 G is soluble and we can consider {G r | r ∈ π(G)}, a Sylow basis of G, where G r ∈ Syl r (G) for each r ∈ π(G). Let q be the largest prime in π(G). For every p ∈ π(G), p = 2, q, by Lemma 2.4 we have that G q is a normal NR-subgroup of the NR-group G q G p G 2 . Then every normal subgroup of G q is normal in G q G p G 2 for every p = 2, q, and so it is normal in G. This implies that G q is an NR-subgroup of G. Moreover G/G q ∼ = G q < G, where G q is a Hall q -subgroup of G. Hence G q is an NR-group and so G/G q is an NR-group. Therefore G is also an NR-group by Lemma 2.2, since G q is a normal NR-subgroup of G. This ends the proof.

Then every second maximal subgroup of G is a soluble T -group (in particular it is a soluble PST-group).
Proof where n is an odd number. By Theorem 2.1 M is a soluble T -group. If r is a prime, then N of type (b) is a T -group. Assume that r = 2 p or 3 p . From the structure of N of type (b) it is a minimal non-abelian group. Clearly, A 4 is also a minimal non-abelian group. This ends the proof.
(4) Clearly, G contains a unique element of order 2, |Z (G)| = 2 and G/Z (G) ∼ = PSL (2, p). Hence (G) = Z (G) and Z (G) < M for every maximal subgroup M of G (see also ([18], Corollary, p. 80)). Therefore by ( [18], 3.6.17, 3.6.25-3.6.26) a maximal subgroup M of G is one of the following groups: (a) Q 8 x where x is an element of order 3, which acts on Q 8 permuting the three maximal subgroups of Q 8 ; (b) Q M, Q ∈ Syl p (G), |Q| = p and M/Q is a cyclic group whose order is relatively prime to p; (c) M = x, y | x n = y 2 , y −1 x y = x −1 and M/Z (G) is a dihedral group of order p ± 1.
The group of type (a) is a minimal non-T -group (see also [16]), so every maximal subgroup of it is a soluble T -group. Clearly the groups of types (b)-(c) are soluble T -groups (the Sylow 2-subgroups of the group of type (c) are either cyclic of order 4 or quaternion groups of order 8). This ends the proof.

Lemma 4.4
Assume that G is one of the following groups: Let M be the group of type (c) and let L be a maximal subgroup of M such that Z (G) < L. From the structure of N it is easily seen that L is abelian. This ends the proof.
Theorem 4.5 Let G be a non-abelian simple group all of whose second maximal subgroups of even order are soluble PST-groups. Then G is one of the following groups: 2. PSL (2, p), where p is a prime with p > 3, p 2 − 1 ≡ 0 (mod 5) and p ≡ 3 or 5 (mod 8); 3. PSL(2, 3 f ), where f is an odd prime and 3 f ≡ 3 (mod 8).
Proof By Lemmas 4.3-4.4 every group of type (1)-(3) is a group all of whose second maximal subgroups of even order are soluble PST-groups.
We will show that there are no other groups satisfying these conditions. Let G be a non-abelian simple group all of whose second maximal subgroups of even order are soluble PST-groups. Let M be an arbitrary maximal subgroup of G. Since soluble PST-groups are NR-groups, by Theorem 4.1 M is either 2-nilpotent or minimal nonnilpotent, in particular M is soluble by Theorem 4.1 and Feit-Thompson Theorem. Then G is a minimal simple group (see Thompson's classification of minimal simple groups in ( [10], Bemerkung II.7.5, p. 190)) and also G is one of the simple groups in Lemma 2.8. Since, if f is odd, then 3 f ≡ 3 (mod 8), this ends the proof. Theorem 4.6 Let G be a non-abelian simple group all of whose second maximal subgroups are soluble PST-groups. Then G is one of the following groups: Proof By Lemma 4.3 every group of type (1)- (3) is a group all of whose second maximal subgroups are soluble PST-groups. Let G be a group all of whose second maximal subgroups are soluble PST-groups. By Theorem 4.5 we should only show that G PSL(2, 3 f ), where f is an odd prime, 3 f ≡ 3 (mod 8) and (3 f − 1)/2 is composite. If not, then G possesses a Frobenius group N with kernel P of order 3 f and a cyclic complement D of order (3 f − 1)/2. For the structure of N see ( [18], p. 393) (in fact, in this notation N ∼ = H/Z (L)). Moreover P is an elementary abelian 3-group. Since (3 f − 1)/2 is composite, it follows that N possesses a proper subgroup x P, where x is a proper subgroup of D of prime order. By hypothesis x P is a soluble PST-group. Then x acts on P as a power automorphism. Let y ∈ P. Then y x P. Hence x y = x × y and so x ∈ C N (y). Since C N (y) P by ([18], 3.6.4(i)), we get a contradiction. This ends the proof. Theorem 4.7 Let G be a group all of whose second maximal subgroups of even order are soluble PST-groups. Then G is either a soluble group or one of the following groups: Proof By Lemmas 4.3, 4.4 every group of type (1)-(5) is a group all of whose second maximal subgroups of even order are soluble PST-groups. We will show that there are no other groups satisfying these conditions. Let G be a group all of whose second maximal subgroups of even order are soluble PST-groups. Assume that G is a non-soluble group. As in the Proof of Theorem 4.5 maximal subgroups of G are either 2-nilpotent or minimal non-nilpotent. Therefore G is one of the groups from Theorem 2.8.
If not, we have G/Z (G) ∼ = PSL (2, p). The rest of the proof is similar to that of (ii) and (v). (vii) G SL(2, 3 f ), where f is even or composite and 3 f ≡ 3 or 5 (mod 8).
Since f is odd, it follows that 3 f ≡ 3 (mod 8). If not, let p be a prime dividing f . By Dickson's Theorem PSL(2, 2 f ) contains a non-soluble proper subgroup PSL(2, 2 p ), a contradiction. From (i)-(x) we obtain the groups of types (1)- (5). This ends the proof.
Theorem 4.8 Let G be a group all of whose second maximal subgroups are soluble PST-groups. Then G is either a soluble group or one of the following groups: