On two classes of generalised finite T-groups

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The springboards for the results of this paper are the classical characterisations of Gaschütz and Robinson of the class of all T -groups, or groups in which every subnormal subgroup is normal, and later developments of them.
Gaschütz proved the following characterisation of soluble T -groups in terms of their normal structure.
Theorem 1 [1,Theorem 2.1.11], [2] Let G be a group and let L be the nilpotent residual of G. Then, G is a soluble T -group if, and only if, the following conditions hold: 1. L is a normal abelian Hall subgroup of G with odd order; 2. G/L is a Dedekind group; 3. Every subgroup of L is normal in G.
Following Robinson (see [1, Definition 2.2.1]), we say that a group G satisfies the property C p ( p a prime) if every subgroup of a Sylow p-subgroup P of G is normal in the normaliser of N G (P). Robinson proved:

Theorem 2 [1, Theorem 2.2.2], [3] A group G is a soluble T -group if, and only if, G is a C p -group for all primes p.
Kegel (see [4,Definition 6.1.4]) introduced an extension of the subnormality to formation theory that emerges naturally from the structural study of the groups.
Recall that if F is a formation, a class of groups which is closed under taking epimorphic images and subdirect products, then every group G has a smallest normal subgroup with quotient in F. This subgroup is called the F-residual of G and it is denoted by G F . We say that the formation F is subgroup-closed if H F is contained in G F for all subgroups H of a group G. Definition 1 Let F be a formation. A subgroup U of a group G is called K-F-subnormal in G if either U = G or there is a chain of subgroups U = U 0 < U 1 < · · · < U n = G such that U i−1 is either normal in U i or U i−1 is a maximal subgroup of U i with U F i ⊆ U i−1 , for i = 1, 2, . . . , n.
It is rather clear that the K-N-subnormal subgroups of a group G, for the formation N of all nilpotent groups, are exactly the subnormal subgroups of G. The reader is referred to [4,Chapter 6] for a full discussion of the properties of these subgroups.
Bearing in mind Kegel's extension of the subnormality within the framework of formation theory and its strong impact on the subgroup structure, it seems interesting to study possible extensions of Gaschütz and Robinson's theorems to this context in order to help us to better understand the role of the subnormality in the subgroup lattice.
It is clear that T N = T . The following theorem confirms that a Gaschütz-type characterization of soluble T F -groups holds.

Theorem 3 [5, Theorem 3]
A group G is a soluble T F -group if, and only if, the following conditions hold: 1. G F is a normal abelian Hall subgroup of G with odd order; 2. X /X F is a Dedekind group for every X ≤ G; 3. Every subgroup of G F is normal in G.
We would like to point up that every T F -group G is in fact a T -group because every subnormal subgroup of G is K-F-subnormal in G.
On the other hand, Skiba [6] generalised the notions of solubility, nilpotency and subnormality introducing the interesting concepts of σ -solubility, σ -nilpotency, and σ -subnormality, in which σ is a partition of the set P, the set of all primes. Thus, P = i∈I σ i , with σ i ∩σ j = ∅ for all i = j.
A group G is said to be σ -primary if the prime factors, if any, of its order all belong to the same member of σ .
If σ is the minimal partition, that is, σ = {{2}, {3}, {5}, . . .}, then the class of σ -soluble groups is just the class of soluble groups and the class of σ -nilpotent groups is just the class of nilpotent groups. Furthermore, the class N σ of all σ -nilpotent groups is a subgroup-closed saturated Fitting formation ([6, Corollary 2.4 and Lemma 2.5]).
Skiba (see [6, Theorem B] and [7]) proved that σ -soluble groups have a good arithmetic behaviour with respect to the members of the partition σ .
where, for every i = 1, . . . , n, H i−1 is normal in The σ -subnormal subgroups play an important role in the study of soluble groups, and they are exactly the subnormal subgroups for the minimal partition. Moreover, the σ -subnormal subgroups of a σ -soluble group G are exactly the K-N σ -subnormal subgroups of G, and so they are a sublattice of the subgroup lattice of G. Furthermore, by [4, Lemma 6.1.6], σsubnormality is a transitive subgroup embedding property, and epimorphic images preserve σ -subnormal subgroups.
Zhang et al. [8] extended the class of all T -groups introducing the class of T σ -groups.
They showed σ -versions of Gaschütz and Robinson's local characterisation of T -groups. To this end, they considered the following extension of the class of all C p -groups. Note that if G satisfies condition C i , then G satisfies condition C p for all p ∈ σ i and every Hall σ i -subgroup of G is Dedekind.
The importance of the property C i is underscored in the next result, which is a characterisation of σ -soluble T σ -groups.
Theorem 5 [8, Theorem 1.10] Let G be a σ -soluble group. Let D = G N σ . Then, the following statements are equivalent. It is important to stress that the σ -soluble T σ -groups are just the T F -groups for the subgroup-closed formation F = N σ of all σ -nilpotent groups. Moreover, every T σ -group is a T -group.
The key to help us to understand the difference between σ -soluble T σ -groups and T -groups is the following theorem.

Theorem A A σ -soluble group G is a T σ -group if, and only if, G is a soluble T -group and the Hall σ i -subgroups of G are Dedekind for all i ∈ I .
We would like to point out that Theorem 5 is a direct consequence of Theorem A. An interesting and smart extension of normality to saturated formations in the soluble universe was given by Doerk and was introduced and studied for the first time in [9]. The definition depends on the minimal local definition of the saturated formation, and can be also considered as a subgroup embedding property in the general finite universe. Definition 7 [9, Definition 3.1] Let F be a non-empty saturated formation and let G be a group. A subgroup A of G is called F-Dnormal in G if A satisfies the following two conditions: The F-Dnormal subgroups turn out to be important in the study of the structure of the groups (see [9,10]).
We are interested here in the N σ -Dnormality. In order to keep our notation homogeneous, Note that the smallest local definition of N σ is given by where E σ i denotes the class of all σ i -groups.
As it was noted in [10, Remark 3.6], if X and H are subgroups of a group G and π is a set of primes, then [X , O π (H )] ≤ H if, and only if, X ≤ N G (O π (H )). Therefore, the next useful characterisation of σ -normal subgroups holds.

Proposition 6 Let A be a subgroup of G and p a prime number. Then, A is a σ -normal subgroup of G if, and only if, the following statements hold:
It is clear that every normal subgroup is σ -normal, and if σ is the minimal partition, then every σ -normal subgroup is normal.
Furthermore, arguing as in [11,Lemma 1.4], it follows that σ -normality is inherited in These facts will be used in the sequel without further reference.
The above basic properties of σ -normal subgroups yield the following.
). Therefore, for every p ∈ π(N ), there exists a Sylow p-subgroup of N normalising O σ i (A). In particular, N normalises O σ i (A) and then O σ i (A) is normal in G. Since A is core-free in G, it follows that O σ i (A) = 1 and so G is a σ i -group. In this case, A is obviously σ -subnormal in G.
The following class of groups naturally emerges from the above proposition.
Note that every T σ -group is in fact a σ T-group. However, these two classes of groups are different because there are σ T-groups which are not T -groups. For instance, if G denotes the alternating group of degree 4, and σ = {{2, 3}, {2, 3} }, then G is an example of a σ T-group which is not a T σ -group since it is not a T -group. Furthermore, the good behaviour of the σ -normal and σ -subnormal subgroups with respect to epimorphic images implies that the class of all σ T-groups is closed by taking quotients.
We point up that the class of all σ T-groups is just the class of all T -groups when σ is the minimal partition.
We say that a group G is a σ -Dedekind group if every subgroup of G is σ -normal in G. It is clear that the class of all σ -Dedekind groups is a subclass of the class of all σ T-groups.
The second main result of the paper is a σ -version of Theorem 1 characterising the σ -soluble σ T-groups. Let G be a σ -soluble group and D = G N σ . Then, G is a σ T-group if, and only if, G satisfies the following statements:

G = D M, where D is an abelian Hall subgroup of odd order;
2. if p ∈ π(D) ∩ σ i , for some i ∈ I , then D contains a Hall σ i -subgroup of G; 3. M is a σ -Dedekind group; 4. every element of G induces a power automorphism of D.
The following generalisation of supersolubility is due to Guo, Chi and Skiba ([12]) Definition 9 A group G is called σ -supersoluble if every chief factor of G below the σnilpotent residual of G is cyclic.
Of course σ -supersolubility coincides with supersolubility for the minimal partition σ . An important consequence of Theorem 1 is that the class of all soluble T -groups is a subgroup-closed class of supersoluble groups. For σ T-groups, the following extension holds.

Proofs of Theorem A and Theorem 5 revisited
The proof of Theorem A depends on the following.

Lemma 9 Let G be a σ -soluble T σ -group. Then, G is soluble.
Proof We argue by induction on |G|. Since the class of all σ -soluble T σ -groups is closed under taking homomorphic images by [4, Lemma 6.1.6] and soluble groups are a saturated formation, we may assume that G is a primitive group with a unique minimal normal subgroup, N say, and G/N soluble. Since G is σ -soluble, it follows that N is σ -primary. Let A be a subgroup of prime order of N . Then, A is σ -subnormal in N . By [4, Lemma 6.1.6], A is σ -subnormal in G. Since G is a T σ -group, we have that A is normal in G. The minimality of N yields N = A and N is abelian. Consequently, G is soluble.

Proof of Theorem A
Assume that G is a σ -soluble T σ -group. Then, by Lemma 9, G is a soluble T -group. By Theorem 3, every σ -nilpotent subgroup of G is a Dedekind group. Consequently, every Hall σ i -subgroup of G is Dedekind for all i ∈ I .
Conversely, assume that G is a soluble T -group and the Hall σ i -subgroups of G are Dedekind for all i ∈ I . Let Z be the σ -nilpotent residual of G. Then, G/Z is a direct product of Hall σ i -subgroups of G/Z , i ∈ I , which are Dedekind. In particular, G/Z is a direct product of nilpotent subgroups and then G/Z is nilpotent. Consequently, Z is actually the nilpotent residual of G. By Theorems 1 and 3, G is a T σ -group.
We now apply Theorem A to give an alternative shorter proof of Theorem 5.

Proof of Theorem 5
If G is a σ -soluble T σ -group and i ∈ I , then the Hall σ i -subgroups of G are Dedekind by Theorem A. Assume that π ⊆ σ i and let H be a Hall π-subgroup of G. Then, H is also a Dedekind group and every subgroup K of H is subnormal in N G (H ). Since  N G (H ) is a T -group, it follows that K is normal in N G (H ). Hence, G satisfies condition C i and Statement (1) implies Statement (2).
Assume now that Statement (2) holds. Then, G satisfies C p for all p. Then, G is a soluble T -group by Theorem 2, and every Hall σ i -subgroup of G is Dedekind. By Theorem A, G is a σ -soluble T σ -group and Statement (2) implies Statement (1).

Clearly, Statement (1) implies Statement (3) because every soluble T -group with Dedekind
Hall σ i -subgroups for all i ∈ I satisfies Statement (3) (note that in this case the nilpotent and σ -nilpotent residuals of G coincide, and we can apply Theorem 1). Assume now that Statement (3) holds. Then, D is just the nilpotent residual of G because M is a Dedekind group. Therefore, by Theorem 1, G is a soluble T -group. Note that if H is a Hall σ i -subgroup of G, then H = O σ i (D) × X for a subgroup X of H . Since H is a T -group and X is nilpotent, it follows that X is a Dedekind group. Note that O σ i (D) is also Dedekind. Since O σ i (D) is of odd order, it follows that H is a Dedekind group. By Theorem A, G is a T σ -group and Statement (1) holds.

The proof of Theorem B
We start by noting that if A is a σ -normal subgroup of a group G and A is a σ i -group for some i ∈ I , then every Sylow p-subgroup of G, for every prime p / ∈ σ i , normalises A. Therefore, . This fact will be also used in the sequel without further reference.

Proof of Theorem B
Assume that G is a σ -soluble σ T-group and let D be the σ -nilpotent residual of G. We prove that G satisfies Statements (1), (2), (3)

and (4) by induction on |G|.
We split the proof into the following steps.
Step 1. D is nilpotent. Assume that D is not nilpotent. If L is a minimal normal subgroup of G, we have that G/L is a σ T-group and DL/L is the σ -nilpotent residual of G/L. The minimal choice of G implies that DL/L is nilpotent. Since the class of all nilpotent groups is a saturated formation, it follows that G has a unique minimal normal subgroup, A say. Furthermore, A is contained in D, D/A is nilpotent and A ∩ (G) = 1. Since G is σ -soluble, it follows that A is a σ i -group for some i ∈ I .
Let B be a subgroup of A. N G (B). This means that A is acted on by conjugation by D as a power automorphisms. In particular, A is abelian, C G (A) = A = F(G), and D is supersoluble. Let p be the largest prime dividing |D|. Then, O p (D) is a Sylow p-subgroup of D and it is a normal subgroup of G. Therefore, Let P be a Sylow p-subgroup of G. Then, X = D P is σ -subnormal in G and so X is a σ T-group. If G = D P, then every subgroup of A ∩ Z(P) = 1 would be normal in G and so A would be cyclic of prime order. In this case A would be central in D and D would be nilpotent, contrary to supposition. Hence, X is a proper subgroup of G. Then, by induction, the σ -nilpotent residual X * of X is an abelian normal Hall subgroup of X contained in D.
Since A is self-centralising in X , it follows that X * is a p-group and X * is a Sylow p-subgroup of G. Therefore, D P = D. In particular, A = P is the unique Sylow p-subgroup of G.
Note that Z = O σ i (G) normalises every subgroup of A. Therefore, Z is a proper subgroup of G. Since Z is a σ T-group and contains D, we have that the σ -residual Z * of Z is normal in G and contained in A. Therefore, Clearly, D = AM * . Assume that q ∈ σ i ∩ π(D) and let Q be a Sylow q-subgroup of D. Then, AQ is a subnormal σ i -subgroup of G. Hence, Q is σ -subnormal in G. Since G is a σ T-group, it follows that Q is σ -normal in G. Thus, D normalises Q and so Q ≤ F(D) = A. Hence, q = p and σ i ∩ π(D) = {p}. This implies that AT is a Hall σ i -subgroup of G and M * is a Hall σ i -subgroup of G.
Observe that D is a proper subgroup of G because G is σ -soluble. Let X be a proper subgroup of G containing D. Then X is σ -subnormal in G and so X is a σ T-group. The induction hypotheses guarantee that X * is an abelian Hall subgroup of X and every subgroup of X * is normal in X . If X * = 1, then X is σ -nilpotent and so is D. Then D is a σ i -group because C G (A) = A and so D = A. Hence D is nilpotent. We may assume that X * = 1. Then X * ∩ A = 1 and so A ≤ X * because A is a Sylow p-subgroup of G. Consequently D * = X * = A. Applying Maschke's Theorem [13,A;Theorem 11.5], and [13, B; Proposition 9.3], A is a direct product of G-isomorphic cyclic minimal normal subgroups of X . Let A 0 one of them. Then C X (A 0 ) = C X (A) = A and X /A is cyclic of order dividing p − 1. Hence every Hall p -subgroup of X is cyclic.
Let S be a subgroup of T of prime order. Assume E = DS is a proper subgroup of G. Then E * = A, and then AS is a σ -subnormal subgroup of E which is σ -subnormal in G. Since AS is a σ i -group, we have that S is σ -subnormal in G. Since G is a σ T-group, it follows that S is σ -normal in G and so A normalises S. This contradiction yields T = 1 and G = D, and this is a contradiction.
Assume G = DS. If C D (S) = 1, then there exists a prime r and a Sylow r -subgroup R of D such that C R (S) = 1. Assume that r = p. Then C A (S) is a normal subgroup of G. Since Step 2. D is a Hall subgroup of G. We may assume that D = 1. Let N be a minimal normal subgroup of G contained in D. Since G/N is a σ T-group, we can apply induction to conclude that D/N is a Hall subgroup of G. Hence (|D/N |, |G/N : D/N |) = 1. Now, N is a p-group for some prime p because D is nilpotent. Assume that p ∈ π(D/N ). Then, (|D|, |G : D|) = 1 and D is a Hall subgroup of G. Consequently, we may assume that N is a Sylow p-subgroup of D.
Let Assume that p ∈ σ i for some i ∈ I . Let G i be a Hall σ i -subgroup of G. Then, and Z is normal in G. This implies that |D| = p.
Clearly, we may assume that G i ∩ M = 1. Let L be a minimal normal subgroup of G contained in G i ∩ M. Let P be a Sylow p-subgroup of G. Then, by induction, DL/L is a Sylow p-subgroup of G/L and so P L = DL and P = D ×(L ∩ P). Note that P is σ -normal in G.
Let p be a prime and let G p be a Sylow p-subgroup of G such that G p ∩ H is a Sylow p-subgroup of H . If p ∈ π(D), then G p normalises H . Suppose that p ∈ π(M), then there exists d ∈ D such that G d p ≤ M, then either G d p normalises H or G d p normalises O σ j (H ). Hence, either G p normalises H d = H or G p normalises (O σ j (H )) d = O σ j (H ). Consequently, H is a σ -normal subgroup of G.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. These results are a part of the R+D+i project supported by the Grant PGC2018-095140-B-I00, funded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way of making Europe". The research of the first author is also supported by the Grant 12071092, funded by the National Science Foundation of China. The authors thank the referees for their careful reading of the paper.
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Conflict of interest
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