A criterion for a normal subgroup to be hypercentral based on class sizes

Ricerche di Matematica https://doi.org/10.1007/s11587-024-00856-7 A criterion for a normal subgroup to be hypercentral based on class sizes Antonio Beltrán1 Received: 6 November 2023 / Accepted: 12 March 2024 © The Author(s) 2024 Abstract Let Gbe a finite group and Na normal subgroup of G. We prove that the knowledge of the sizes of the conjugacy classes of Gthat are contained in Nand of their multiplicities provides information of Nin relation to the structure of G. Among other results, we obtain a criterion to determine whether a Sylow p-subgroup of Nlies in the hypercentre of Gfor a fixed prime p, and therefore, whether the whole subgroup Nis hypercentral in G. Keywords Conjugacy class sizes ·Normal subgroups ·Nilpotent groups · Hypercentre Mathematics Subject Classification 20D15 ·20E45 1 Introduction It is well known that the set of conjugacy class sizes of a finite group may provide structural information of it. There exist multiple relevant results that endorse it, for instance, Itô’s theorem about the nilpotency of groups having exactly two class sizes. The reader is referred to [3] for a good survey on this widely studied research topic. But the knowledge of only the class sizes of a group may result quite restrictive to get to know certain properties of it. In fact, the order of the group is unbounded, and its nilpotency or solvability cannot be determined either ([4] and [10], respectively). Nevertheless, when the multiplicities of the class sizes are provided as well, it seems that further information of the group could be obtained. Indeed, a reference work by Cossey, Hawkes and Mann [5] establishes that nilpotency can be read off from the knowledge of the class sizes and their multiplicities, however, recognising solvability and supersolvability of a group still continue to be open problems. BAntonio Beltrán [email protected] 1Departamento de Matemáticas, Universitat Jaume I, 12071 Castellón, Spain 123 A. Beltrán Let Gbe a finite group and NG. Instead of looking into all conjugacy classes of G, we restrict ourselves to the conjugacy classes of Gcontained in N, which are called G-classes of N. There also exist research works (see [1]or[2] for instance) showing that a limited set of G-class sizes of Nmay exert a strong influence on the structure of N. Our aim in this paper is to investigate which properties of Ncan be extracted from the G-class sizes together with their multiplicities. For this purpose, inspired by the class size frequency function introduced in [5], we define the G-class size frequency function of N,wG N:N−→ N, as follows: wG N(n)=1 n|{g∈N:|gG|=n}|, that is, for every n∈N,wG N(n)is the number of G-classes contained in Nthat have cardinality n. This function trivially provides arithmetical properties of N; its order and hence, the order of its Sylow subgroups, as well as the number of elements of N that are central in G. No information of G, whenever N<G, seems to be able to be recovered. Even though the G-class sizes and their multiplicities may involve primes that do not divide the order of N, we show, however, that the G-class size frequency function of Ncan still provide properties of Nwith regard to its immersion in G.Our first result is Theorem A Let G be a finite group and N a normal subgroup of G. Then the G-class sizes of N and their multiplicities determine whether N is hypercentral in G. By using properties of the hypercentre of a group, Theorem A yields to the following corollary, which shows how the G-class size frequency function of a normal subgroup Ngives information about the external behaviour of the prime-power order elements of Nwith respect to G. Corollary B Let G be a finite group, N a normal subgroup and p a prime. Then the G-class sizes of N and their multiplicities determine whether all p-elements of N belong to CN(Op(G)). Inspired by [9], we obtain the next result relative to commutativity in Gof certain prime-power order elements of N. Theorem C Suppose that G is a finite group and N is a normal subgroup of G. Then the G-class sizes of N and their multiplicities determine whether all elements of N lying in the centre of some Sylow subgroup of G belong to Z(G). We will show by an example that the G-class size frequency function of a normal subgroup does not detect whether such subgroup is abelian, nor the derived length when the normal subgroup is solvable or the nilpotency class when it is nilpotent. We have been unable to decide, however, whether nilpotency can be recognised. Of course, determining solvability or supersolvability, as happens with class sizes and their multiplicities in a group, are likewise open problems. We encourage the reader to continue with this research. 123 A criterion for a normal subgroup to be... 2 Notation and preliminaries Let Gbe a finite group and Na normal subgroup. For our convenience, if πis a set of primes, we will write SG π(N)to denote the subset of elements of Nwhose G-class has cardinality a π-number. The cardinality of this set is easily computed from the G-class size frequency function of N,wG N, defined in the Introduction, by the formula |SG π(N)|=  nπ=1 wG N(n)·n, where nπdenotes the π-part of n∈N. We will specially focus on the cases π= {p}and π={p}with pprime, and will write SG p(N)and SG p(N)to denote the corresponding sets. We recall the definition of the hypercentre of a group [8,p.7,8].LetGbe a finite group and 1 ≤Z1(G)≤Z2(G)≤...be a series of subgroups of G, where Z1(G)is the centre of Gand Zi+1(G)for i≥1 is defined by Zi+1(G)/Zi(G)=Z(G/Zi(G)). Then the hypercentre of G,Z∞(G), is the last term of this series, and is nilpotent and characteristic in G. A group Gis nilpotent if and only if G=Z∞(G). Before giving the proofs of our results, we need to state and develop several basic lemmas about hypercentres and normal subgroups. The first property is renowned and the second, which originally appeared in a slightly modified form in [11, Lemma 2], is a consequence of it, which is often more convenient to apply. Lemma 2.1 (p. 739, [7]) Let G be a finite group. A normal p-subgroup N of G lies in Z∞(G)if and only if G/CG(N)is a p-group. Lemma 2.2 (Lemma 2, [5]) A p-element x of a finite group G lies in Z∞(G)if and only if x commutes with each p-element of G. It is well known that the primes dividing |Z∞(G)|are exactly those dividing |Z(G)|. The same occurs when one intersects with any normal subgroup of G. Lemma 2.3 Let G be a finite group and N a normal subgroup of G. If a prime p divides |N∩Z∞(G)|, then p divides |Z(G)∩N|. Proof Suppose that pdivides |N∩Z∞(G)|and take P0a Sylow p-subgroup of N∩Z∞(G). Since N∩Z∞(G)is nilpotent and normal in G,wehaveP0G.Now, if Pis a Sylow p-subgroup of G, it is clear that P0≤P, and hence P0∩Z(P)= 1. By applying Lemma (2.1)toP0, we get G=PCG(P0)and this implies that 1 = P0∩Z(P)≤Z(G). Since we also have P0∩Z(P)≤N, we conclude that pdivides |N∩Z(G)|. 3 Proofs We are ready to prove our results. We remark that our proof of Theorem A, which we state again in a more convenient form, differs from that originally appeared in [5] 123 A. Beltrán to demonstrate that the nilpotency of a group can be recognised from the class size frequency function. We have adapted instead the ideas of [8, Theorem 23.5]. Theorem 3.1 Let G be a finite group and N a normal subgroup of G. Then the G-class sizes of N and their multiplicities determine whether N is hypercentral in G. Precisely, if p is a prime number, then (a) |SG p(N)|p=|N∩Z∞(G)|p, where Z∞(G)is the hypercentre of G. (b) a Sylow p-subgroup of N is hypercentral in G if and only if |SG p(N)|p=|N|p. Proof We proceed by induction on |N|to prove (a). By definition of the set SG p(N) we have |SG p(N)|≡|N∩Z(G)|(mod p). Suppose first that pdoes not divide |N∩Z∞(G)|, so certainly pdoes not divide |N∩Z(G)|either. In this case the above congruence proves that |SG p(N)|is a p- number. Hence |SG p(N)|p=1=|N∩Z∞(G)|p, so (a) is proved. Henceforth, we will assume that pdivides |N∩Z∞(G)|. By Lemma (2.3), we have that pdivides |N∩Z(G)|, so we can take M≤N∩Z(G)such that |M|=p.We consider G=G/Mwith the normal subgroup N=N/M. In the following we will prove |SG p(N)|·p=|SG p(N)|(3.1) For every g∈N,wehave CG(g)/M≤CG(g)=U. If u∈U, then we can write gu=gf(u), with f(u)=[g,u]∈M.AsMis central in G, then f(uv) =[g,uv]=[g,v][g,u]v=[g,u][g,v]= f(u)f(v), for all v∈U. Thus, fis a group homomorphism from Uto M, and its kernel is exactly CG(g). Consequently, |U/CG(g)|divides |M|=p. Take now g∈SG p(N), that is, take g∈Nsatisfying |gG|=pifor some i∈N. By the above paragraph we know that either U=CG(g)or |U|=p|CG(g)|.Inthe former case we have |gG|=|G:CG(g)|=|G:CG(g)|=pi, and in the latter, |gG|=|G:CG(g)|=p|G:U|=p|G:CG(g)|=pi+1. We consider the set of G-classes of Nof cardinality pi, whose number is given by the G-class frequency function, wG N(pi), and the correspondence, by taking bars, 123 A criterion for a normal subgroup to be... between G-classes of Nand G-classes of N. Next we see that every G-class of N, with regard to pre-images, gives rise to G-classes of Nin two different ways. Write M=z(with zof order p) and let g∈SG p(N). We have seen above that either |gG|=|gG|or p|gG|=|gG|. In the first case, CG(g)=CG(g), and this easily leads to that gG,(gz)G,...,(gzp−1)Gare pdistinct G-classes of Nthat are pre-images of gG, with the same cardinality as gG.Ifp|gG|=|gG|, then the inverse image of gGis exactly the G-class of N,gG, with cardinality p|gG|. Accordingly, for every i≥0, we can decompose wG N(pi)into two summands wG N(pi)=w1(pi)+w2(pi), where w1(pi)and w2(pi)denote the number of G-classes of Ncorresponding to each one of both types of classes. Therefore |SG p(N)|= ∞  i=1 piw1(pi−1)+ ∞  i=0 pipw2(pi) = ∞  i=1 piw1(pi−1)+ ∞  i=1 piw2(pi−1) =p∞  i=1 (pi−1w1(pi−1)+pi−1w2(pi−1) =p ∞  i=1 pi−1wG N(pi−1)=p|SG p(N)|, so Eq. (3.1) is proved. As M≤Z(G), then the definition of the hypercentre implies that Z∞(G)= Z∞(G). Hence Z∞(G)∩N=(Z∞(G)∩N)/M. Then, we utilize the inductive hypothesis and Eq. (3.1) to get |Z∞(G)∩N|p=p|Z∞(G)∩N|p=p|SG p(N)|p=|SG p(N)|p, and the proof of (a) is finished. For proving (b), notice first that |SG p(N)|p=|N|pimplies by (a) that |Z∞(G)∩ N|p=|N|p. Then, it trivially follows that a Sylow p-subgroup of Nis hypercentral in G. Conversely, if a Sylow p-subgroup Pof Nlies in Z∞(G), then |P|=|N|p≤|Z∞(G)∩N|p, so |N|p=|Z∞(G)∩N|p. By applying (a) we obtain the equality |SG p(N)|p=|N|p, so(b)isproved. Now, Nis hypercentral in Gif and only if every Sylow subgroup of Nis hypercentral in G, and this can be established by (b) because |SG p(N)|pand |N|pare given by the 123 A. Beltrán G-class size frequency function of N. Therefore, the first assertion of the theorem is proved.  Remark. Observe that |N∩Z(G)|=wG N(1)and likewise, by Theorem 3.1(a), |N∩ Z∞(G)|can be retrieved from the G-class frequency function of N. We stress that, however, it is not possible in general to compute the intermediate terms |N∩Zi(G)| when i>1. This follows straightforwardly from [5, Example 1]. Nonetheless, we will give an easier example at the end of this section. On the other hand, we note that Theorem 3.1(a) shows that a Sylow p-subgroup of Nis hypercentral in Gif and only if |SG p(N)|pis as large as possible. Proof of Corollary B By Theorem 3.1(b), we know that the G-class size frequency function of Ndetermines whether the Sylow p-subgroups of Nlie in Z∞(G). In fact, when this occurs then Nhas only one Sylow p-subgroup. Thus, by Lemma (2.2), this function detects whether all p-elements of Ncentralize each p-element of G. Since Op(G) is generated by all p-elements of G, the result follows.  We state again Theorem C in a more convenient manner. Theorem 3.2 Suppose that G is a finite group and N is a normal subgroup of G. Then the G-class sizes of N and their multiplicities determine whether all elements of N lying in the centre of some Sylow subgroup of G belong to Z(G). More precisely, if p is a prime number, then (a) |N∩Z(G)|divides |SG p(N)|. (b) |CN(Op(G))|divides |SG p(N)|. (c) |SG p(N)|p=|N∩Z(G)|pif and only if Z(P)∩N≤Z(G), where P is any Sylow p-subgroup of G. Proof To prove (a), we notice that an element x∈Nhas G-class size a p-number if and only if CG(x)contains a Sylow p-subgroup of G. Thus, SG p(N)= P∈Sylp(G) CN(P). Since N∩Z(G)⊆CN(P)for every P∈Sylp(G), we conclude that SG p(N)is a union of cosets of N∩Z(G), whence we deduce (a). As Op(G)coincides with the subgroup of Ggenerated by all Sylow p-subgroups of G,wehave CN(Op(G)) = P∈Sylp(G) CN(P)⊆SG p(N). It follows that SG p(N)is a union of cosets of CN(Op(G)), and this proves (b). For proving (c), we fix a Sylow p-subgroup Pof Gand put Z:= P∩N∩Z(G), which is a Sylow p-subgroup of N∩Z(G)and normal in G. Next, we claim that if n∈SG p(N)then CG/Z(nZ)=CG(n)/Z. Set U/Z:= CG/Z(nZ).Itisclear that CG(n)≤U. Furthermore, if u∈U, then we can write nu=nf(u)for some 123 A criterion for a normal subgroup to be... f(u)∈Z. Since Z≤Z(G), identically as in the proof of Theorem 3.1,itfollows that fis a group homomorphism from Uto Z, whose kernel is exactly CG(n).Asa consequence, U/CG(n)is isomorphic to a subgroup of Z. However, Zis a p-group whereas |U/CG(n)|divides |nG|, which is a p-number. This forces U=CG(n),so the claim is proved. We observe now that SG p(N)is also a union of cosets of Zand consider the set ¯ S of such cosets and the group P/Zacting by conjugation on ¯ S. By using the claim it easily follows that a coset nZ ∈¯ Sis fixed by P/Zif and only if nis fixed by P,or equivalently, nZ ∈CN(P)/Z. Now, by applying the orbit formula to this action we get |¯ S|=|SG p(N)|/|Z|≡|CN(P)|/|Z|(mod p). On the other hand, since |Z|=|N∩Z(G)|pwe have that the equality |SG p(N)|p= |Z|=|N∩Z(G)|pholds if and only if |SG p(N)|/|Z|is a p-number, and this is equivalent to |CN(P)|/|Z|being a p-number. Now, as Z(P)is a Sylow p-subgroup of CG(P), then Z(P)∩Nis a Sylow p-subgroup of CN(P). Thus, we deduce that |CN(P)|/|Z|is a p-number if and only if Z=Z(P)∩N. Now it is easily seen that this condition is equivalent to the fact that Z(P)∩N≤Z(G), and therefore, the proof of (c) is finished. Now, since for every prime p, the numbers |SG p(N)|pand |N∩Z(G)|pare provided by the G-class size frequency function of N, the first assertion of the theorem follows.  Example An easy example illustrates that, as indicated in the Introduction, the fact that a normal subgroup is abelian cannot be read off from the set of its G-class sizes and their multiplicities. Indeed the same group may possess two normal subgroups showing this. Let D16 =x,y|x8=y2=1,xy=x−1and D8be the dihedral groups of order 16 and 8 respectively. Set G=D16 ×D8, or equivalently G=SmallGroup(128,2011) in the library of small groups within GAP [6], and let us consider its normal subgroups xand D8.ThesetofG-class sizes of both subgroups, counting multiplicities, is {1,1,2,2,2}, whilst xis abelian and D8is not. This example also serves to show, as pointed out in the Remark, that from the G-class size frequency function of a normal subgroup Nwe cannot compute the sizes of all the terms N∩Zi(G)for every i. Indeed, Z2(G)=x2×D8,so|x∩Z2(G)|=4 and |D8∩Z2(G)|=8. In view of the above example, we can affirm that the derived length of a solvable normal subgroup as well as the nilpotency class of a nilpotent one are not determined from the G-class size frequency function. Acknowledgements This work is supported by Proyecto CIAICO/2021/193, Generalitat Valenciana, Spain. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Declarations Conflict of interest The author declares that he has no financial interests. 123 A. Beltrán Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Akhlaghi, Z., Beltrán, A., Felipe, M.J.: Normal sections, class sizes and solvability of finite groups. J. Algebra 399, 220–231 (2014) 2. Alemany, E., Beltrán, A., Felipe, M.J.: Nilpotency of normal subgroups having two G-class sizes. Proc.Amer.Math.Soc.139(8), 663–669 (2011) 3. Camina, A.R., Camina, R.D.: The influence of class sizes on the structure of finite groups: a survey. Asian-Eur. J. Math. 4(4), 559–588 (2011) 4. Camina, A.R., Camina, R.D.: Recognising nilpotent groups. J. Algebra 300, 16–24 (2006) 5. Cossey, J., Hawkes, T., Mann, A.: A criterion for a group to be nilpotent. Bull. London Math. Soc. 24, 267–270 (1992) 6. The GAP Group, GAP - Groups, Algorithms and Programming, Vers. 4.4.12 . http://www.gap-system. org(2008) 7. Huppert, B.: Endliche Gruppen. Springer-Verlag, Berlin (1967) 8. Huppert, B.: Character Theory of Finite Groups. Walter de Gruyter, Berlin (1998) 9. Mattarei, S.: Retrieving information about a group form its character degrees or from its class sizes. Proc.Amer.Math.Soc.143(8), 2189–2195 (2006) 10. Navarro, G.: The set of conjugacy class sizes of a finite group does not determine its solvability. J. Algebra 411, 47–49 (2014) 11. Peng, T.A.: The hypercenter of a finite group. J. Algebra 48, 46–56 (1977) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123