FINITE GROUPS WITH HEREDITARILY G-PERMUTABLE SCHMIDT SUBGROUPS

Abstract A subgroup A of a group G is said to be hereditarily G-permutable with a subgroup B of G, if 
$AB^x = B^xA$
 for some element 
$x \in \langle A, B \rangle $
 . A subgroup A of a group G is said to be hereditarily G-permutable in G if A is hereditarily G-permutable with every subgroup of G. In this paper, we investigate the structure of a finite group G with all its Schmidt subgroups hereditarily G-permutable.


Introduction
All groups considered in the paper are finite.
Recall that a group G is said to be a minimal nonnilpotent group or Schmidt group if G is not nilpotent and every proper subgroup of G is nilpotent.It is clear that every nonnilpotent group contains Schmidt subgroups, and their embedding has a strong structural impact (see, for example, [2,3,10]).
However, the following extensions of permutability turn out to be important in the structural study of groups and were introduced by Guo et al. in [6].DEFINITION 1.1.Let A and B be subgroups of a group G.
(1) A is said to be G-permutable with B if there exists some g ∈ G such that AB g = B g A. (2) A is said to be hereditarily G-permutable with B (or G-h-permutable with B, for short) if there exists some g ∈ A, B such that AB g = B g A. [2] Hereditarily G-permutable Schmidt subgroups 523 (3) A is said to be G-permutable in G if A is G-permutable with all subgroups of G.
(4) A is said to be hereditarily G-permutable (or G-h-permutable, for short) in G if A is hereditarily G-permutable with all subgroups of G.
It is clear that permutability implies G-permutability but the converse does not hold in general as the Sylow 2-subgroups of the symmetric group of degree 3 show.
Our main goal here is to complete the structural study of groups in which every Schmidt subgroup of a group G is G-h-permutable.This study was started in [2] where we prove the following important fact.
Observe that the alternating group of degree 4 is a nonsupersoluble Schmidt group.Let p 1 > p 2 > • • • > p r be the primes dividing |G| and let P i be a Sylow p i -subgroup of G, for each i = 1, 2, . . ., r.Then we say that G is a Sylow tower group of supersoluble type if all subgroups P 1 , P 1 P 2 , . . ., P 1 P 2 • • • P r−1 are normal in G.The class of all Sylow tower groups of supersoluble type is denoted by D.
Recall that if F is a nonempty class of groups and π is a set of primes, then F π is the class of all π-groups in F. In particular, if p is a prime, then N p is the class of all p-groups and D π(p−1) is the class of all Sylow tower groups G of supersoluble type such that every prime dividing |G| also divides p − 1.
If G is a group, then Soc(G) is the product of all minimal normal subgroups of G and Φ(G) is the Frattini subgroup of G, that is, the intersection of all maximal subgroups of G.
Our main goal here is to describe completely the groups G with trivial Frattini subgroup which have their Schmidt subgroups G-h-permutable.THEOREM 1.3.Let G be a group with Φ(G) = 1.Assume that F = LF(F) is the saturated formation locally defined by the canonical local definition F such that F(p) = N p D π(p−1) for every prime p.If every Schmidt subgroup of G is G-h-permutable in G, then the following statements hold: We shall adhere to the notation and terminology of [1,4].

Definitions and preliminary results
Our first lemma collects some basic properties of G-h-permutable subgroups which are very useful in induction arguments.Its proof is straightforward.LEMMA 2.1.Let A, B and K be subgroups of G with K normal in G.Then, the following statements hold.
The following result describes the structure of Schmidt groups.LEMMA 2.2 [5,8].Let S be a Schmidt group.Then S satisfies the following properties: (1) the order of S is divisible by exactly two prime numbers p and q; (2) S is a semidirect product S = [P] a , where P is a normal Sylow p-subgroup of S and a is a nonnormal Sylow q-subgroup of S and a q ∈ Z(S); (3) P is the nilpotent residual of S, that is, the smallest normal subgroup of S with nilpotent quotient; (4) P/Φ(P) is a noncentral chief factor of S and Φ(P) = P ⊆ Z(S); ( 5) Φ(S) = Z(S) = P × a q ; (6) Φ(P) is the centraliser C P (a) of a in P; (7) if Z(S) = 1, then |S| = p m q, where m is the order of p modulo q.
In what follows, Sch(G) denotes the set of all Schmidt subgroups of a group G. Following [3], a Schmidt group with a normal Sylow p-subgroup will be called an S p,q -group.
The proof of Theorem 1.3 follows after a series of lemmas.They give us an interesting picture of the groups with supersoluble Schmidt subgroups.Let q ∈ π(G) with q p and let Q be a Sylow q-subgroup of G. Since N = C G (N), it follows that PQ is not nilpotent.Hence, G has an S p,q -subgroup S, which is supersoluble p-closed because G ∈ F.Then, by statements (4) and ( 5) of Lemma 2.2, |S/ Z(S)| = pq and therefore, by statement (7) of Lemma 2.2, q divides p − 1.Since G is a Sylow tower group of supersoluble type, it follows that and thus G ∈ H, which is a contradiction.Hence, F ⊆ H.
Assume that F H, and let G be a group in H \ F of minimal order.Since H is a saturated formation and F(p) is a formation of soluble groups for all primes p, it follows that G is a primitive soluble group.Let N be a unique minimal normal subgroup of G.The choice of G yields G ∈ H and G/N ∈ F. Since G is soluble, N is a p-group for some prime p, and from G ∈ H, it follows that [4,Lemma A.13.6].
Let S be an S r,q -subgroup of G.If r p, then S is contained in some Hall , it follows that q divides p − 1.Thus, by Lemma 2.2, S ∈ U. Consequently, every Schmidt subgroup of G is supersoluble, which is a contradiction.Hence, F = H.
The following examples show that groups in Lemma 2.3 may not be supersoluble.
The following result is of interest although it is not needed for the proof of Theorem 1.3.PROPOSITION 2.6.Let F = {H | Sch(H) ⊆ U}.Then, for every n ∈ N, there exists a group G ∈ F of nilpotent length n.
Assume that G 1 is a cyclic group of order p 1 .Assume that i ≥ 2 and G i−1 is in F and of nilpotent length i − 1.By [4,Corollary B.11.8], G i−1 has a faithful and irreducible module V p i over the field of p i elements.Let G i = [V p i ]G i−1 be the corresponding semidirect product.Then F(G i ) = V p i and hence the nilpotent length of G i is equal to i. Furthermore, by Lemma 2.3, G i ∈ F. In particular, G n is an F-group of nilpotent length n.
The following subgroup embedding property was introduced by Vasil'ev, Vasil'eva and Tyutyanov in [9].DEFINITION 2.7.A subgroup H of a group G is said to be P-subnormal in G if there exists a chain of subgroups Note that P-subnormality coincides with K-U-subnormality (see [1,Ch. 6]) in the soluble universe.LEMMA 2.8.Let A be a G-h-permutable subgroup of a soluble group G.Then, A is P-subnormal in G.In particular, the supersoluble residual A U of A is subnormal in G.
PROOF.Let G be a group of smallest order for which the lemma is not true, and let L be a minimal normal subgroup of G. Since G is soluble, |L| = p n for some prime p ∈ π(G) and n ≥ 1. Suppose that G = AL.Then A is a maximal subgroup of G and A ∩ L = 1.Let L 1 be a subgroup of prime order of L.Then, AL , we conclude that |G : A| = p and then A is P-subnormal in G, which is a contradiction.Hence, G AL. Since |AL| < |G|, by Lemma 2.1, it follows that A is P-subnormal in AL.By Lemma 2.1, AL/L is (G/L)-h-permutable in G/L, and from |G/L| < |G|, it follows that AL/L is P-subnormal in G/L.In particular, AL is P-subnormal in G by [1,Lemma 6.1.6].However, then A is a P-subnormal subgroup of G by [1, Lemma 6.1.7],which is a contradiction.Consequently, A is P-subnormal in G. Applying [1, Lemma 6.1.9],we conclude that A U is subnormal in G. EXAMPLE 2.9.Let G be a group isomorphic to the alternating group of degree 6.Since G does not have maximal subgroups of prime index, the identity subgroup 1 of G is G-h-permutable but not P-subnormal in G. Thus, the solubility of the group G in Lemma 2.8 is essential.

LEMMA 2 . 3 .
Let F = {H | Sch(H) ⊆ U}, where U is the class of all supersoluble groups.Then, F satisfies the following properties:(1) if G ∈ F, then G is a Sylow tower group of supersoluble type; in particular, G is a soluble group; (2) F is a subgroup-closed saturated Fitting formation; (3) U ⊆ F; (4) F = LF(F), where F is the canonical local definition such that F(p) = N p D π(p−1)for every prime p.PROOF.Statements (1), (2) and (3) follow from [7, Lemma 4 and Theorem 2].Let H = LF(F) be a local formation defined by the formation function F with F(p) = N p D π(p−1) for every prime p. Assume that F H. Let G be a group in F \ H of minimal order.Since F is a saturated formation, it follows that G is a primitive soluble group.Let N = Soc(G) be the unique minimal normal subgroup of G. Then G/N ∈ H. Since G is a Sylow tower group of supersoluble type and C G (N) = N, we see that N is a Sylow p-subgroup of G, where p is the largest prime dividing |G|.

EXAMPLE 2 .
4. Let Q = a, b | a 4 = b 4 = 1, a 2 = b 2 , b −1 ab = a −1 be the quaternion group of order 8. Then G has a faithful and irreducible module A over the field of 5 elements of dimension 2. Let G = [A]Q be the corresponding semidirect product.Then G is not supersoluble and C = [A] a and D = [A] b are supersoluble and normal subgroups of G = CD.By Lemma 2.3, G ∈ F = {H | Sch(H) ⊆ U}.EXAMPLE 2.5.Assume that M is a nonabelian group of order 21.Then M has a faithful and irreducible module N over GF(43), the field of 43 elements (see, for example, [4, Corollary B.11.8]).Consider the semidirect product