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On Groups Whose Non-Normal Subgroups Are Either Contranormal Or Core-Free * 1 Introduction

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On Groups Whose Non-Normal Subgroups Are Either Contranormal Or Core-Free * 1 Introduction Advances in Group Theory and Applications c 2020 AGTA - www.advgrouptheory.com/journal 10 (2020), pp. 83–125 ISSN: 2499-1287 DOI: 10.32037/agta-2020-011 On Groups whose Non-Normal Subgroups are either Contranormal or Core-Free * L.A. Kurdachenko —A.A.Pypka —I.Ya.Subbotin (Received Feb. 6, 2020;AcceptedFeb.27, 2020 —CommunicatedbyF.deGiovanni) Abstract AsubgroupH of a group G is called contranormal in G if HG = G.AsubgroupH of a group G is called core-free in G if CoreG(H)= 1 .Obviously,thesetwotypes of subgroups are the complete opposite of normal subgroups.h i In this paper, we will obtain the structure of soluble and non-soluble groups whose non-normal subgroups are contranormal. Moreover, we will obtain the structure of some periodic groups whose non-normal subgroups are either contranormal or core-free. Mathematics Subject Classification (2020): 20E34, 20F16, 20F50 Keywords:contranormalsubgroup;core-freesubgroup 1 Introduction Let G be a group. The following two normal subgroups are associ- ated with any subgroup H of G: HG,thenormal closure of H in G, the least normal subgroup of G that contains H,andCoreG(H),the normal core of H in G,thegreatestnormalsubgroupofG which is contained in a subgroup H. We have HG = Hx| x G h 2 i * The first two authors are supported by the National Research Foundation of Ukraine (project 2020.02/0066) 84 L.A. Kurdachenko – A.A. Pypka – I.Ya. Subbotin and x CoreG(H)= H . x G \2 AsubgroupH is normal in G if and only if CoreG(H)=H.Inthis sense, the subgroups H,forwhichCore (H)= 1 ,arethecomplete G h i opposite to the normal subgroups. A subgroup H of a group G is called core-free in G if Core (H)= 1 . G h i AsubgroupH is normal in G if and only if HG = H.Inthissense, the subgroups H,forwhichHG = G,arethecompleteoppositesto the normal subgroups. A subgroup H of a group G is called contranor- mal in G if HG = G.J.S.Rosehasintroducedtheterm“contranormal subgroup” in the paper [20]. For each subgroup H of a group G we have the following two extreme and opposite situations: HG = H or HG = G, and, respectively, Core (H)=H or Core (H)= 1 . G G h i The following extreme cases immediately appear. The first case: every proper subgroup of G is normal. Such group is called a De- dekind group.AdescriptionofDedekindgroupshasbeenobtained by R. Baer [1]. The second case: every proper subgroup of G is core- free. In this case, G does not contain proper non-trivial normal sub- groups, that is, G is a simple group. The third case: every proper non-trivial subgroup of G is contranormal. In this case, G does not contain proper non-trivial normal subgroups, so that again G is a simple group. In the last two cases we came only to simple groups. Note again that a simple group has the only three following types of subgroups: normal, core-free and contranormal. Therefore, the following question natural appears: what we can say about the groups whose subgroups are either normal, core-free or contranor- mal? We study groups whose subgroups are either normal or core-free, and the groups whose subgroups are either normal or contranormal. Note that groups having only two types of subgroups, which are also antagonistic in some sense to each other, have been considered by many authors. Here we provide a list of papers whose subject is to some extent related to our topic: [5, 6, 7, 8, 11, 14, 16, 17, 18, 22]. On groups with normal, contranormal or core-free subgroups 85 Groups whose subgroups are either normal or core-free have been studied in the paper [15]. The study of groups whose subgroups are either normal or contranormal was initiated in the paper [22]. In The- orem 2 of [22], basis structural features of such groups were shown. However, this theorem was not provided with a proof. Therefore, in this paper we provide a more detailed description of such groups. AgroupG is called quasisimple,ifthecentralfactor-groupG/⇣(G) is simple and G =[G, G]. It is not hard to show that every subgroup of quasisimple group is either normal or contranormal. Theorem A Let G be a group whose non-normal subgroups are contra- normal. If G is not soluble, then G is simple or quasisimple. Recall that an infinite generalized quaternion group is a group Q = A b h i where A is a normal Prüfer 2-group,1 |b| = 4, b2 A and ab = a-1 2 for all a A. 2 Theorem B Let G be a soluble group whose non-normal subgroups are contranormal. Suppose that G is not a Dedekind group. If G is a p-group for some prime p,thenp = 2 and G is a group of one of the following types: (i) G = D g where D is a normal divisible abelian 2-subgroup, g has h h i order 2 or 4, dg = d-1 for every d D; 2 (ii) G = D g where D is a normal divisible abelian 2-subgroup, g has h i order 2 or 4, dg = d-1 for every d D, D = A B,whereA is 2 ⇥ aPrüfer2-subgroup, g2 ⌦ (A) and A, g is an infinite general- 2 1 h i ized quaternion group. If G is a periodic group and |⇧(G)| > 2,thenG is a group of one of the following types: (iii) G = S h g where g is a p-element for some prime p, S is an h i p abelian Sylow p0-subgroup of G, C g (S)= g and every subgroup h i h i of S is G-invariant; (iv) G = D g where D is a normal abelian subgroup, D = S K h h i ⇥ where S is a Sylow 20-subgroup of G, K is a divisible 2-subgroup, g has order 2 or 4, dg = d-1 for every d D; 2 86 L.A. Kurdachenko – A.A. Pypka – I.Ya. Subbotin (v) G = D g where D is a normal abelian subgroup, D = S A B h i ⇥ ⇥ where S is a Sylow 2 -subgroup of G, A B is a divisible 2-sub- 0 ⇥ group, g has order 2 or 4, dg = d-1 for every d D, g2 ⌦ (A), 2 2 1 and A, g is an infinite generalized quaternion group. h i If G is a non-periodic group, then G is a group of one of the following types: g -1 (vi) G = D h g , g has order 2 or 4,andx = x for each x D, h i 2 2 2 C g (D)= g , 2 ⇧(D), D = D,andeverysubgroupofD h i h i 62 is G-invariant; g -1 (vii) G = D h g , g has order 2 or 4,andx = x for each x D, h i2 2 2 C g (D)= g , D = D, D = S B, 2 ⇧(B), S is a divisible Sy- h i h i ⇥ 62 low 2-subgroup of D; (viii) G = D g , g has order 2 or 4,andxg = x-1 for each x D, h i 2 2 2 C g (D)= g , D = D, D = S B,where2 ⇧(B), S = A C is h i h i ⇥ 62 ⇥ adivisibleSylow2-subgroup of D, g (A B)= 1 , g2 ⌦ (C) h i\ ⇥ h i 2 1 and C, g is an infinite generalized quaternion group. h i Further, for the study of groups whose subgroups are either nor- mal, core-free or contranormal, it is natural to assume that they con- tain proper contranormal and proper non-trivial core-free subgroups. Here we consider periodic locally soluble such groups. Their descrip- tion decomposes to few natural parts. Theorem C Let G be a group whose non-normal subgroups are either contranormal or core-free. If G is locally soluble, then G is a soluble group. Let G be a group and A be a normal subgroup of G.Thein- tersection MonG(A) of all non-trivial G-invariant subgroups of A is called the G-monolith of A.IfMonG(A) is not trivial, then A is called G-monolithic.IfA = G,thenwewillsaythatMonG(G) is the monolith of G and denote it by Mon(G). Recall that a p-group G is called extraspecial,if[G, G]=⇣(G) is a subgroup of order p and G/⇣(G) is an elementary abelian p-group. The following theorem is dedicated to the monolithic case. Theorem D Let G be a soluble periodic monolithic group whose non-nor- mal subgroups are either contranormal or core-free. Suppose that G contains proper contranormal and non-trivial core-free subgroups. Then G is a group of one of the following types: On groups with normal, contranormal or core-free subgroups 87 (i) G = D v where D is a normal Prüfer 2-subgroup, v2 = 1, h h i dv = d-1 for all d D; 2 (ii) G = M h S where M is an elementary abelian p-subgroup, p is a prime, S is a locally cyclic p0-subgroup, CG(M)=M,andevery complement to M in G is conjugate with S;inparticular,ifM is finite, then G is finite and G = M h S where S is a cyclic Sylow p0-subgroup of G; (iii) G = D g where D is a normal cyclic p-subgroup, |g| = q is a h h i prime, q<p, CG(D)=D; (iv) G = D g where D is a normal Prüfer p-subgroup, |g| = q is a h h i prime, q<p, CG(D)=D; (v) G = D g where D is an extraspecial p-subgroup, p is a prime, h h i |g| = q is a prime, q<p, q = 2,moreover,M =[D, D]=⇣(D) is a 6 monolith of G,andeverysubgroupofD/M is G-invariant; (vi) G = M h K where M is a finite elementary abelian p-subgroup, p is an odd prime, K is a quaternion group of order 8, M is a minimal normal subgroup of G, CG(M)=M; (vii) G = M h B where M is a minimal normal elementary abelian p-sub- group, p is an odd prime, B = K u where K is a normal Prü- h h i fer 2-subgroup, u2 = 1, au = a-1 for each a K; 2 (viii) G = M h B where M is a minimal normal elementary abelian p-sub- group, p is an odd prime, B is an infinite generalized quaternion group; (ix) G = M h V where M is a minimal normal elementary abelian p-sub- group, p is a prime, V = D g where D is a locally cyclic p -sub- 1 h h i 1 0 group, |g| = p,everysubgroupofD is g -invariant, C (D )=D ; 1 h i V 1 1 (x) G = M h V where M is a minimal normal elementary abelian p-sub- group, p is a prime, V = D g where D is a locally cyclic sub- 1 h h i 1 group, g is a q-element, q is an odd prime, p, q ⇧(D1), q 62 C g (D1)= g ,andeverysubgroupofD1 is g -invariant; h i h i h i (xi) G = M h V where M is a minimal normal elementary abelian p-sub- group, p is a prime, V = D1 h g where D1 is a locally cyclic sub- h i 2 g -1 group, g is a 2-element, 2, p ⇧(D1), C g (D1)= g , x = x 62 h i h i for each x D ; 2 1 88 L.A.
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