A subgroup H of a group G is said to be quasinormal if HX =
XH for all subgroups X of G. In this article groups are characterized for
which the partially ordered set of quasinormal subgroups is decomposable

de Falco, M.

de Giovanni, F.

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 27 (2001), 137-142GROUPS WITH DECOMPOSABLE SET OF QUASINORMALSUBGROUPSM. De Falco, F. de Giovanni, C. MusellaCommunicated by V. DrenskyAbstract. A subgroup H of a group G is said to be quasinormal if HX =XH for all subgroups X of G. In this article groups are characterized forwhich the partially ordered set of quasinormal subgroups is decomposable.1. Introduction. A partially ordered set L is said to be decompos-able if it is isomorphic to the direct product of two non-trivial partially orderedsets. The structure of groups for which a certain relevant system X of subgroupsis decomposable has been investigated for several choices of the system X. Inparticular, about fifty years ago Suzuki [7] proved that the lattice L(G) of allsubgroups of a group G is decomposable if and only if G is the direct product oftwo periodic coprime non-trivial subgroups. Groups with decomposable latticeof normal subgroups have been later described by Curzio [2], and more recentlydecomposition problems for the ordered sets of subnormal, ascendant or othergeneralized normal subgroups have been studied (see [3], [4], [5]).A subgroup H of a group G is said to be quasinormal (or permutable) ifHX = XH for every subgroup X of G. Clearly the join of any set of quasinor-mal subgroups is likewise quasinormal, but the intersection of two quasinormal2000 Mathematics Subject Classification: 20E15.Key words: quasinormal subgroup, decomposable ordered set.138 M. De Falco, F. de Giovanni, C. Musellasubgroups of a group need not be quasinormal. The aim of this article is to char-acterize groups G for which the partially ordered set qn(G) of all quasinormalsubgroups is decomposable.Let G be a group, and let H be a subgroup of G. The H-norm of G isthe subgroup ker(G : H) consisting of all elements g of G such that Xg = X foreach subgroup X of G containing H, i.e.ker(G : H) =⋂H≤XNG(X).Clearly, H ≤ ker(G : H) ≤ NG(H), and ker(G : H) ≤ ker(G : K) ifH ≤ K ≤ G.Moreover, the {1}-norm ker(G : {1}) is the norm N(G) of G, introduced by Baer[1]; it is well-known that the subgroup N(G) is contained in the second term ofthe upper central series of G.Theorem. Let G be a group. The partially ordered set qn(G) of allquasinormal subgroups of G is decomposable if and only if G = G1 ×G2, whereG1 and G2 are non-trivial subgroups of G satisfying the following conditions:(a) every quasinormal subgroup of Gi is quasinormal in G (for i = 1, 2);(b) if H1 and H2 are quasinormal subgroups of G1 and G2, respectively, thefactor groups ker(G1 : H1)/H1 and ker(G2 : H2)/H2 have no elements ofthe same prime order.It should be observed that condition (a) in the above statement is nottrivial, since quasinormal subgroups of direct factors need not be quasinormal inthe whole group. To show this, let H be a finite p-group (p prime) containing aquasinormal non-normal subgroup X, and let x and h be elements of X and H,respectively, such that h−1xh /∈ X; if h has order pn and 〈z〉 is a cyclic group oforder pn, then X is not quasinormal into the direct product G = H × 〈z〉, sinceX〈hz〉 6= 〈hz〉X.Most of our notation is standard; in particular, if H and K are subgroupsof a group G with H ≤ K, we shall denote by [K/H] the set of all subgroups Xof G such that H ≤ X ≤ K. We will use the monograph [6] as a general referenceon quasinormal subgroups and lattice properties of groups.2. Proof of the Theorem.Lemma 1. Let G be a group, and let H be a quasinormal subgroup of G.Then every subgroup of ker(G : H) containing H is quasinormal in G.Groups with decomposable set of quasinormal subgroups 139P r o o f. Let X be a subgroup of G such thatH ≤ X ≤ ker(G : H),and let Y be any subgroup of G. Since X is contained in the normalizer ofHY , it follows that the product XY = X(HY ) is a subgroup of G. ThereforeXY = Y X, and X is a quasinormal subgroup of G. Lemma 2. Let the group G = G1 × G2 be the direct product of twosubgroups G1 and G2, and let H be a subgroup of G1 which is quasinormal in G.If g is an element of ker(G1 : H) such that the subgroup 〈g,H〉 is quasinormal inG and the index |〈g,H〉 : H| is a prime number, then g belongs to ker(G : H).P r o o f. Let X be any subgroup of G containing H, and assume bycontradiction that Xg 6= X. Put K = 〈g,H〉 and |〈g,H〉 : H| = p, so that gp liesin X, and there exists an element x of X such that the conjugate xg does notbelong to 〈x〉H. Write x = x1x2, with x1 ∈ G1 and x2 ∈ G2. Obviously〈x〉〈x1〉H = 〈x〉H ≤ 〈x〉H,and hence g /∈ 〈x1〉H. It follows thatH ≤ K ∩ 〈x1〉H < K,and so K ∩ 〈x1〉H = H. Since the subgroup K is quasinormal in G, we havexg = gmhxn, where h is an element of H and m,n are integers. Thus(x1g)x2 = xg = gmhxn = (gmhxn1 )xn2 ,and hence x1g = gmhxn1and x2 = xn2. On the other hand, g belongs to thesubgroup ker(G1 : H), and so xg1= xr1h1, where r is an integer and h1 is anelement of H. It follows that gmhxn1= gxr1h1, and u = gm−1h = xr1h1x−n1belongs to K ∩ 〈x1〉H = H. Thereforexg = xg1x2 = xr1h1x2 = uxn1x2 = uxnis an element of 〈x〉H, and this contradiction proves the lemma. Let G be a group, and let X be any subgroup of G. Since every join ofquasinormal subgroups is likewise quasinormal, we can define the quasinormalcore of X (in G) as the largest quasinormal subgroup of G which is containedin X. In particular, if H and K are quasinormal subgroups of a group G, thequasinormal core of H ∩ K in G will be denoted by H ∧ K; then H ∩ K isquasinormal in G if and only if H ∩ K = H ∧ K. Note also that the partially140 M. De Falco, F. de Giovanni, C. Musellaordered set qn(G) is a lattice, but it is not in general a sublattice of the subgrouplattice L(G) of G.P r o o f o f t h e Th e o r em. Suppose first that the ordered set qn(G) isdecomposable, and letϕ : qn(G) −→ L1 × L2be an order isomorphism, where L1 and L2 are non-trivial partially ordered sets.Clearly Li has smallest element Oi and largest element Ii (i = 1, 2), and thepreimages G1 = ϕ−1(I1, O2) and G2 = ϕ−1(O1, I2) are quasinormal subgroups ofG such that G1G2 = G and G1 ∧G2 = {1}. If a1 and a2 are elements of L1 andL2, respectively, we have(a1, a2) = sup{(a1, O2), (O1, a2)}and(a1, O2) = inf{(a1, a2), (I1, O2)}, (O1, a2) = inf{(a1, a2), (O1, I2)}.It follows that, if H is any quasinormal subgroup of G, thenH = (H ∧G1)(H ∧G2),and so alsoH = 〈H ∩G1,H ∩G2〉.Let g be any element of G1; then Gg2is a quasinormal subgroup of G, and henceGg2= (Gg2∧G1)(Gg2∧G2) = (G2 ∧G1)g(Gg2∧G2) = Gg2∧G2,so that Gg2≤ G2 and G2 is normal in G = G1G2. A similar argument shows thatalso G1 is a normal subgroup of G; then G1 ∩G2 is normal in G, and soG1 ∩G2 = G1 ∧G2 = {1}.Therefore G = G1 × G2 is the direct product of its non-trivial subgroups G1and G2. Let X be any quasinormal subgroup of G1. Then the subgroup XG2 isquasinormal in G, and henceXG2 = (XG2 ∧G1)(XG2 ∧G2) = (XG2 ∧G1)G2.As XG2 ∧G1 is contained in X, it follows that X = XG2 ∧G1 is a quasinormalsubgroup of G. It can be proved similarly that every quasinormal subgroupof G2 is also quasinormal in G. Let H1 and H2 be quasinormal subgroups ofG1 and G2, respectively, and assume by contradiction that the factor groupsGroups with decomposable set of quasinormal subgroups 141ker(G1 : H1)/H1 and ker(G2 : H2)/H2 contain cyclic subgroupsK1/H1 = 〈g1H1〉and K2/H2 = 〈g2H2〉, respectively, with the same prime order p. It follows fromLemma 1 that K1 and K2 are quasinormal in G1 and G2, respectively. ThenH1,H2,K1,K2 are quasinormal subgroups of G, so that also H = H1H2 andK = K1K2 are quasinormal in G. Moreover, H is normal in K andK/H = 〈g1H〉 × 〈g2H〉is an elementary abelian group of order p2. Let L be any subgroup of G suchthat H < L < K, and put L/H = 〈gm1gn2H〉. Application of Lemma 2 yields thatg1 ∈ ker(G : H1) and g2 ∈ ker(G : H2), so that the subgroup 〈g1, g2〉 is containedin ker(G : H), and L = 〈gm1gn2,H〉 is quasinormal in G by Lemma 1. Thereforethe interval [K/H] of the lattice L(G) is contained in qn(G), and it follows imme-diately that the lattice L(K/H) ≃ [K/H] is decomposable. This contradictionproves that ker(G1 : H1)/H1 and ker(G2 : H2)/H2 have no elements of the sameprime order.Conversely, suppose that the group G = G1 ×G2 satisfies the conditionsof the statement, and let H be any quasinormal subgroup of G. Put H1 = H∩G1and H2 = H ∩ G2, and assume that the subgroup H1H2 is properly containedin H. Consider an element h of H \ H1H2, and write h = x1x2 with x1 ∈ G1and x2 ∈ G2. Thus x1 ∈ K1 \H1 and x2 ∈ K2 \H2, where K1 = HG2 ∩G1 andK2 = HG1 ∩G2. Let X be any subgroup of G1 containing H1; thenXK1 ≤ XHG2 = XH ≤ XH,and henceXK1 ≤ XH ∩G1 = XH1 = X.Therefore K1 is contained in ker(G1 : H1). A similar argument shows that K2 isa subgroup of ker(G2 : H2). For i = 1, 2 the subgroup Hi is quasinormal in Gi,and Lemma 1 yields that every subgroup of Ki containing Hi is also quasinormalin Gi. It follows now from the hypotheses that the factor groups K1/H1 andK2/H2 are periodic withpi(K1/H1) ∩ pi(K2/H2) = ∅.Since the non-trivial cosets x1H1 and x2H2 have the same order, we get a con-tradiction. ThereforeH = (H ∩G1)(H ∩G2)for every quasinormal subgroup H of G. As the sets qn(G1) and qn(G2) arecontained in qn(G), the positionHϕ = (H ∩G1,H ∩G2)142 M. De Falco, F. de Giovanni, C. Muselladefines an order isomorphismϕ : qn(G) −→ qn(G1)× qn(G2),and hence the partially ordered set qn(G) is decomposable. REFERENCES[1] R. Baer. Der Kern, eine charakteristische Untergruppe. Compositio Math.1 (1934), 254–283.[2] M. Curzio. Sui gruppi per cui sono riducibili alcuni reticoli di sottogruppinotevoli. Matematiche (Catania) 19 (1964), 1–10.[3] M. De Falco, C. Musella. Some lattice properties of nearly normal sub-groups. Quaderni Mat., 7 (2001), to appear.[4] S. Franciosi, F. de Giovanni. Sui gruppi con l’insieme ordinato dei sot-togruppi ascendenti riducibile. Istit. Veneto Sci. Lett. Arti Atti Cl. Sci. Mat.Natur. 139 (1980/81), 199–202.[5] S. Rinauro. Some lattice properties of virtually normal subgroups. NoteMat. 10 (1990), 273–278.[6] R. Schmidt. Subgroup Lattices of Groups. De Gruyter, Berlin, 1994.[7] M. Suzuki. On the lattice of subgroups of finite groups. Trans. Amer. Math.Soc. 70 (1951), 345–371.Dipartimento di Matematica e ApplicazioniUniversita` di Napoli “Federico II”Complesso Universitario Monte S. AngeloVia CintiaI 80126 NapoliItalye-mails:musella@matna2.dma.unina.itdegiova@matna2.dma.unina.itdefalco@matna2.dma.unina.it Received January 4, 2001

Groups with Decomposable Set of Quasinormal Subgroups

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Groups with Decomposable Set of Quasinormal Subgroups

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