Ukrainian Mathematical Journal, Vol. 65, No. 11, April, 2014 (Ukrainian Original Vol. 65, No. 11, November, 2013)

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ON FINITE GROUPS WITH PERMUTABLE GENERALIZED SUBNORMAL SUBGROUPS

V. F. Velesnitskii and V. N. Semenchuk UDC 512.542

We study the Kegel–Shemetkov problem of finding the classes of finite groups F such that, in any finitegroup, the product of permutable F -subnormal subgroups is a F -subnormal subgroup.

All groups considered in the present paper are finite. By the classic Wieland theorem [1], the set of allsubnormal subgroups forms a lattice in any finite group. In [2], Kegel generalized this result and showed that theset of all F-composition subgroups forms a lattice in any finite group if F is a hereditary formation closed relativeto the extensions.

In the theory of classes of finite groups, the notion of subnormality is a natural generalization of the notion ofF-normality (F-attainability). We now recall these notions.

Let F be a nonempty formation. A subgroup H of the group G is called F-subnormal if either H = G orthere exists a maximum chain

G = H0 ⊃ H1 ⊃ . . . ⊃ Hn = H

such that (Hi−1)F ⊆ Hi for all i = 1, 2, . . . , n.

A somewhat different notion of F-subnormality was introduced by Kegel in [2]. In fact, it combines thenotions of subnormality and F-subnormality.

A subgroup H is called Kegel F-subnormal or F-attainable if there exists a chain of subgroups

G = H0 ⊇ H1 ⊇ . . . ⊇ Hm = H

such that, for any i = 1, 2, . . . ,m, either the subgroup Hi is normal in Hi−1 or (Hi−1)F ⊆ Hi.

At present, F-subnormal (F-attainable) subgroups are called generalized subnormal subgroups.In 1978, Kegel and Shemetkov stated the following problem:

Problem 1 [2, 3]. Determine the classes of groups F such that the set of all generalized subnormal subgroups

of any finite group forms a lattice.

The complete solution of Problem 1 of finding hereditary saturated formations F with the lattice property forF-attainable (F-subnormal) subgroups in the class of soluble groups was obtained by Ballester-Bolinches, Doerk,and Perez-Ramos in [4]. In the general case, this problem is solved by Vasil’ev, Kamornikov, and Semenchuk [5].

Recall that a formation F possesses the lattice property if, for any group G, the subgroups �H,K� and H∩K

are F-subnormal in G for any its F-subnormal subgroups H and K.

Skorina Gomel Sate University, Gomel, Belarus.

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 11, pp. 1555–1559, November, 2013. Original article submitted Novem-ber 20, 2012.

1720 0041-5995/14/6511–1720 c� 2014 Springer Science+Business Media New York

ON FINITE GROUPS WITH PERMUTABLE GENERALIZED SUBNORMAL SUBGROUPS 1721

If the conditions of generation of F-subnormal subgroups are replaced with a weaker condition, namely, withthe product of permutable F-subnormal subgroups, and the condition of saturation of formations F is removed,then Problem 1 can be generalized as follows:

Problem 2. Determine the classes of groups F such that, for any group G and any its permutable F-

subnormal (F-attainable) subgroups H and K, the subgroup HK is F-subnormal (F-attainable) in G.

In the present paper, we determine nonempty hereditary formations F satisfying the conditions formulated inProblem 2. The following theorem is proved:

Theorem 1. Let F be a nonempty hereditary superradical formation. Then, for any group G and any its

permutable F-subnormal subgroups H and K, the subgroup HK is F-supernormal in G.

In what follows, we need the following definitions and notation:

Let π be a set of prime numbers, let Gπ be the class of all π -groups, and let S be the class of all solublegroups. By π

� we denote the complement to π in the set of all prime numbers; if π = {p}, then we write p�

instead of π�.

If F is a class of groups and G is a group, then the coradical GF is the intersection of all normal subgroupsN from G such that G/N ∈ F.

A formation is defined as a class of groups closed with respect to quotient groups and subdirect products.By π(F) we denote the set of all prime numbers p for which there exists a nonidentity p-group in F.

A formation F is called X-superradical if any group G ∈ X such that G = AB, where A,B ∈ F and areF-subnormal in G, belongs to F.

If X is the class of all finite groups, then an X-superradical formation is called superradical.Let F and X be nonempty formations of finite groups. Recall that the product of formations is defined as

follows: FX = {G | GX ∈ F}.If F is a class of groups, then the group G is called a minimum non-F-group if it does not belong to F and

any its proper subgroup belongs to F. The set of all these minimum non-F-groups is denoted by M(F).

We now present the known properties of F-subnormal subgroups.

Lemma 1. Let F be a nonempty hereditary formation. Then the following statements are true:

(i) if H is a subgroup of the group G and GF ⊆ H, then H is a F-subnormal subgroup of the group G;

(ii) if H is a F-subnormal subgroup of the group G, then H ∩K is a F-subnormal subgroup of K for any

subgroup K of the group G.

Lemma 2. Let F be a nonempty formation, let H and N be subgroups of the group G, and let, in addition,

N be normal in G. Then:

(i) if H is F-subnormal in G, then HN is F-subnormal in G and HN/N is F-subnormal in G/N ;

(ii) if N ⊆ H, then H is F-subnormal in G if and only if H/N is F-subnormal in G/N.

Lemma 3 [7]. Let W = Zpn−1 � Zp and let B be an interlacing base, p ∈ P, n ∈ N. Then the following

statements are true:

(i) W contains a subnormal subgroup isomorphic to Zpn ;

(ii) if M = [B,Zp], N = MZp, and ω ∈ N \M, then ωp = 1;

1722 V. F. VELESNITSKII AND V. N. SEMENCHUK

(iii) W = BN, where B and N are normal subgroups of W with exponent pn−1

, n ≥ 2.

Lemma 4. Let F be a nonempty hereditary S-superradical formation. Then Nπ(F) ⊆ F.

Proof. First, we prove that any primary minimum non-F-group is cyclic. Let G ∈ M(F) and let G be ap-group. If G is not cyclic, then, in G, one can find two different maximal subgroups M1 and M2. It is clear thatthey are normal in G and G/Mi ∈ F, Mi ∈ F, i = 1, 2. This implies that GF ⊆ Mi. By Lemma 1, M1 and M2

are F-subnormal subgroups of the group G. Since G = M1M2 and F is a S-superradical formation, G ∈ F,

which is impossible.We now show that Nπ(F) ⊆ F. Assume the contrary. Let G be a group of the least order from Nπ(F) \ F.

Since Nπ(F) is a hereditary formation, G is a minimal non-F-group. We show that G is a primary group. Let��π(G)�� > 1. Since G is nilpotent, we have G = A × B. It is clear that G/A ∈ F and G/B ∈ F. Since F is a

formation,

G � G/A ∩B ∈ F,

which is impossible. Hence, G is a p-group. As shown above, G is a cyclic p-group. Let |G| = pn, where n is

a fixed natural number.If n = 1, then G is a group of prime order p. Since G ∈ Nπ(F), where π(F) is a characteristic of the

formation F, we conclude that G ∈ F, which is impossible.Let n > 1. Consider a group W = Zpn−1 �Zp. Then W = BZp, where B is an interlacing base. By Lemma

3, the group W contains a subgroup P isomorphic to G. Since P ∈ M(F) and F is a hereditary formation, thegroup W does not belong to F.

By Lemma 3, W = BN, where B and N are normal subgroups of the group W with exponent pn−1. Note

that B ∈ F and N ∈ F. This yields W/B ∈ F and W/N ∈ F. Thus, we conclude that

WF ⊆ B ∩N.

According to Lemma 1, B and N are F-subnormal subgroups of the group W. Since F is a S-superradicalformation, we get W ∈ F, which is a contradiction.

The lemma is proved.

Proof of Theorem 1. We prove the theorem by induction on the order of the group G. Let A and B bepermutable F-subnormal subgroups of the group G. Denote T = AB. Assume that N is a minimal normalsubgroup of the group G. In view of Lemma 2, by induction, we show that TN/N is a F-subnormal subgroupof G/N. By Lemma 2, TN is a F-subnormal subgroup of the group G. If TN �= G, then, by induction, T is aF-subnormal subgroup of TN, which means that T is F-subnormal in G.

Now let TN = G for any normal subgroup N of the group G. It is clear that TG = 1. If AF �= 1, then, byvirtue of Lemma 1, the subgroup A

F is F-subnormal in G. Thus, according to Theorem 7.10 in [3],

1 �= (AF)G = (AF)TN ⊆ T.

This means that TG �= 1, which is a contradiction. Hence, AF = 1.

Similarly, we can prove that BF = 1.

We now show that AN ∈ F and BN ∈ F. Consider the following two cases:

ON FINITE GROUPS WITH PERMUTABLE GENERALIZED SUBNORMAL SUBGROUPS 1723

1. Let N be an Abelian subgroup. Since N is a minimum normal subgroup of the group G, we concludethat N is a p-group. We show that p ∈ π(F). Since

AN/N � A/A ∩N

and A ∈ F, we get AN/N ∈ F. This yields (AN)F ⊆ N.

Let (AN)F = N. Since A is a F-subnormal subgroup of the group G, by Lemma 1, A is a F-subnormal subgroup in AN. Since A ∈ F, it is clear that A is a proper subgroup of AN. Then A ⊆ M,

where M is a maximal F-normal subgroup in AN. It is clear that (AN)F ⊆ M. Then

A(AN)F = AN ⊂ M,

which is impossible. Thus, (AN)F ⊂ N. Since AN/(AN)F ∈ F and p ∈ π�AN/(AN)F

�, we conclude

that p ∈ π(F). By Lemma 4, we have Nπ(F) ⊆ F. This implies that N ∈ F. By Lemma 1, N is a F-subnormal subgroup of AN. Since F is a superradical formation, we conclude that AN ∈ F. Similarly,we can show that BN ∈ F.

2. Let N be a non-Abelian subgroup. Then

N = N1 ×N2 × . . .×Nt

is the direct product of isomorphic non-Abelian simple groups. Since A ∈ F, we get AN/N ∈ F. Thisyields (AN)F ⊆ N. If (AN)F = N, then AN = (AN)FA. If A is a proper subgroup of AN, thenA ⊆ M, where M is a maximum F-normal subgroup in AN. Since (AN)F ⊆ M, we have M = AN,

which is impossible. Thus, A = AN and AN ∈ F. Now let (AN)F ⊂ N. If (AN)F �= 1, then

(AN)F = Ni1 ×Ni2 × . . .×Nin .

Since F is a hereditary formation, we conclude that N/(AN)F ∈ F. Thus, it is easy to see thatN ∈ F. By Lemma 1, N is a F-subnormal subgroup of AN. Since F is a superradical formation,we have AN ∈ F. Similarly, we show that BN ∈ F. By Lemma 2, AN and BN are F-subnormalsubgroups of the group G. Since F is a superradical formation, we get G ∈ F. Since F is a maximumformation, T is a subnormal subgroup of the group G.

The theorem is proved.

By I we denote a subset of N × N. Let πi and πj be sets of prime numbers, and let Gπi and Gπj be theclasses of all πi -groups and πj -groups, respectively. In what follows, we consider formations of the form

F =�

(i,j)∈I

GπiGπj .

Recall that a group G is called p-closed (p-nilpotent) if its Sylow p-subgroup (Sylow p-complement) isnormal in G. The group G is called p-decomposable if it is simultaneously p-closed and p-nilpotent. ThenGp�Gp is the class of all p-nilpotent groups, GpGp� is the class of all p-closed groups, Gp�Gp

�GpGp� is the

1724 V. F. VELESNITSKII AND V. N. SEMENCHUK

class of all p-decomposable groups, and N =�

Gp�Gp is the class of all nilpotent groups, where p runs over allprime numbers.

A group G is called π -nilpotent (π -decomposable) if it is p-nilpotent (p-decomposable) for any prime num-ber p from π. The classes of all π -nilpotent (π -decomposable) groups can be represented in the form

�

p∈πGp�Gp

�

p∈π

�Gp�Gp

�GpGp�

�.

A group G is called π -closed if it has a normal π -Hall subgroup. Then GπGπ� is the class of all π -closedgroups.

In [8], it is proved that formations of the form F =�

(i,j)∈I GπiGπj are superradical.Thus, in view of Theorem 1, we arrive at the following result:

Theorem 2. Let F be either the class of all p-closed groups, or the class of all p-nilpotent groups, or

the class of all p-decomposable groups, or the class of all π -nilpotent groups, or the class of all π -decomposable

groups, or the class of all π -closed groups. Then, for any group G and any its permutable F-subnormal subgroups

H and K, the subgroup HK is F-subnormal in G.

It is easy to see that the obtained results remain true if the notion of F-subnormality is replaced by the notionof F-attainability.

REFERENCES

1. H. Wielandt, “Uber den Normalisator der subnormalen Untergruppen,” Math. Z., 69, No. 8, 463–465 (1958).2. O. H. Kegel, “Untergruppenverbande endlicher Gruppen, die Subnormalteilerverband echt enthalten,” Arch. Math., 30, No. 3, 225–

228 (1978).3. L. A. Shemetkov, Formations of Finite Groups [in Russian], Nauka, Moscow (1978).4. A. Ballester-Bolinches, K. Doerk, and M. D. Perez-Ramos, “On the lattice of F -subnormal subgroups,” J. Algebra, 148, No. 2,

42–52 (1992).5. A. F. Vasil’ev, S. F. Kamornikov, and V. N Semenchuk, “On the lattices of subgroups of finite groups,” in: Infinite Groups and

Related Algebraic Systems [in Russian], Kiev (1993), pp. 27–54.6. V. N Semenchuk, “Soluble F -radical formations,” Mat. Zametki, 59, No. 2, 261–266 (1996).7. K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin–New York (1992).8. V. N Semenchuk and L. A. Shemetkov, “Superradical formations,” Dokl. Nats. Akad. Nauk Belarus., 44, No. 5, 24–26 (2000).