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Note on structural properties of graphs

Note on structural properties of graphs 1IntroductionOver the years, graphs generated from a group or a semigroup have been extensively studied. For example, in 1964, Bosak [1] studied graphs induced by semigroups; in [2] the intersection graphs of non-trivial subgroups of abelian groups with finite order were studied by Zelinka; in [3,4,5] the directed graph defined over the elements of a group, the well-known Cayley digraph, was studied; in the books in [6,7,8], numerous valuable applications of this kind of graph are presented, hence the importance of researching them. Another best-known graph is the so-called directed power graph, whose definition was given by Kelarev and Quinn [9], which is defined in such a way that it is possible to apply it also to semigroups, and thus, it was in [10,11,12] that power graphs of semigroups were considered for the first time. These papers use only the short-term “power graph.” However, it makes sense for both directed and undirected power graphs. In addition, Kelarev and Quinn [11] defined an interesting class of directed graphs over semigroups, the semigroup divisibility graphs. There is also one more example, the well-known hyperbolic graphs, mentioned in the works [13,14,15], whose main objects of study were initially Cayley graphs associated with finite groups, nowadays these graphs have several applications in Physics and Geometry. For more information on structural properties associated with a graph, see [16,17,18].It is said that group theory started with Galois (1811–1832), who showed that the best way to understand polynomials is by relating them to certain groups of permutations of their roots. From then onward, group theory has become a useful tool for different fields of mathematics, such as combinatorics, geometry, logic, number theory, and topology.At the end of the nineteenth century, there were two main streams of group theory: on the one hand topological groups (especially Lie groups); on the other hand, finite groups. The latter since its beginnings in the nineteenth century has grown to become an extensive and diverse part of algebra, in particular, the theory of locally finite groups and the theory of nilpotent, solvable groups [19,20]. In the early 1980s, this development culminated in the classification of finite simple groups, impressively and convincingly demonstrating the strength of its methods and results. Thanks to those works, all finite groups that are constructible from these simple groups are now known.Graph theory is an increasing mathematical discipline containing deep and strong results of high applicability. Its rapid development in the last few decades is not only due to its status as the main structure on which currently applied mathematics (computer science, combinatorial optimization, and operations research) is supported, but also due to its increasing connections in the applied sciences.In this paper, we are concerned with the commuting graph GG{{\mathcal{G}}}_{G}of a group GG. It is defined as the graph with GGas the set of vertices and where two of them form an edge if and only if they commute. This graph has been studied from several perspectives, for example, in [21] the chromatic and clique numbers are obtained for the commuting graphs of the dihedral-type groups; in [22] it is proved that the commuting graph of a finite minimal non-solvable group has diameter ≥3\ge 3; in [23] the authors obtain the number of spanning trees of the commuting graph for some specific groups, as well as the classification of finite groups for which the power graph and the commuting graph coincide; and in [24] properties of this graph, as coloring and independence number, are used to prove results about finite groups.The existence of abelian subgroups of maximal order of a given group GGis a topic widely studied in different papers. For example, in [25] it was shown that if mmis the maximal order of an abelian subgroup of a finite group GG, then ∣G∣| G| divides m!m\&#x0021;and in [26] some results are presented on mm. Moreover, in [27] a classification of the abelian subgroups of maximal order of finite irreducible Coxeter groups is obtained, the geometry of these subgroups is studied and some applications of such classification to statistical physics are given.In [24] Bertram, by using the commuting graph GG{{\mathcal{G}}}_{G}, gave an upper bound for the order of an abelian subgroup of a finite group GG. In addition, he proposed to find necessary and sufficient conditions for the existence of an abelian subgroup whose order attains the bound ∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{m}]| . In this paper, we give a solution to this problem and present some related results.In [28] the following result appears.Lemma 1.1If, for some qqthe number of vertices of degree ≥q\ge qis ≤q\le q, then the graph G{\mathcal{G}}can be qq-colored.And it is used in [24] to prove the following theorem.Theorem 1.2Let GGbe a finite group with conjugacy classes indexed by cardinality: 1=∣[x1]∣≤∣[x2]∣≤⋯,1=| \left[{x}_{1}]| \le | \left[{x}_{2}]| \hspace{0.33em}\le \cdots \hspace{0.33em},and let CG(x){C}_{G}\left(x)denote the centralizer of xx. If mmis the smallest number iisatisfying∣[x1]∣+∣[x2]∣+⋯+∣[xi]∣≥∣CG(xi)∣,| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{i}]| \ge | {C}_{G}\left({x}_{i})| ,then each abelian subgroup AAof GGsatisfies(1)∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣≥∣A∣.| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{m}]| \ge | A| .The problem proposed in [24] is the following:Find necessary and sufficient conditions on GGin order that inequality (1) in Theorem 1.2 becomes an equality for some abelian subgroup A<GA\lt G.In this paper, we use the following notation and known facts. Let G{\mathcal{G}}be an undirected simple graph, whose vertex set is V={x1,x2,…,xn}V=\left\{{x}_{1},{x}_{2},\ldots ,{x}_{n}\right\}. The degree of a vertex xxis the number of edges incident with xxand it is denoted by d(x)d\left(x). We say that the vertices of G{\mathcal{G}}can be kk-colored when there exists a partition of VVinto kksubsets, with no two vertices in the same subset connected by an edge of G{\mathcal{G}}. The minimum number kkfor which G{\mathcal{G}}can be kk-colored is called the chromatic number of G{\mathcal{G}}and it is denoted by χ(G)\chi \left({\mathcal{G}}). A complete subgraph of G{\mathcal{G}}is a subgraph when every pair of vertices is connected by an edge. A maximal complete subgraph of G{\mathcal{G}}is called a clique; we also use the term kk-clique for a clique consisting of kkvertices. The clique number ω(G)\omega \left({\mathcal{G}})is the maximal size of a clique contained in G{\mathcal{G}}. Note that if G{\mathcal{G}}can be kk-colored, then the number of vertices in each clique (as well as in each complete subgraph) has cardinality ≤k\le k; hence, we always have ω(G)≤χ(G)\omega \left({\mathcal{G}})\le \chi \left({\mathcal{G}}). A graph G{\mathcal{G}}is called weakly perfect if ω(G)=χ(G)\omega \left({\mathcal{G}})=\chi \left({\mathcal{G}})(see [29]).A subset XXof a finite group GGis called a commuting set if xy=yxxy=yxfor any x,y∈Xx,y\in X. The commuting number κ(G)\kappa \left(G)of a finite group GGis the maximum cardinality of a commuting set. A subset Y⊆GY\subseteq Gis called an anti-commuting set if xy=yxxy=yximplies x=yx=yfor any x,y∈Yx,y\in Y. The Λ\Lambda -number of GGis the minimal number kksuch that GGcan be partitioned into kkanti-commuting subsets; it is denoted by Λ(G)\Lambda \left(G).From the proof of Theorem 1.2, we get the following remark.Remark 1.3The graph GG{{\mathcal{G}}}_{G}is ∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{m}]| -colorable, thus, χ(GG)≤∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣.\chi \left({{\mathcal{G}}}_{G})\le | \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{m}]| .2ResultsThroughout this paper we denote by ccthe sum ∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +| \left[{x}_{m}]| , the left hand side of inequality (1). Note that ω(GG)≤χ(GG)≤c.\omega \left({{\mathcal{G}}}_{G})\le \chi \left({{\mathcal{G}}}_{G})\le c.2.1GG{{\mathcal{G}}}_{G}parameters and the bound ccTheorem 2.1A finite group GGcontains an abelian subgroup of order ccif and only if GG{{\mathcal{G}}}_{G}contains a cc-clique.Corollary 2.2A finite group GGcontains an abelian subgroup AAof order ccif and only if GG{{\mathcal{G}}}_{G}is a weakly perfect graph with χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=c.ProofIf GGcontains an abelian subgroup of order cc, then c≤ω(GG)≤χ(GG)≤cc\le \omega \left({{\mathcal{G}}}_{G})\le \chi \left({{\mathcal{G}}}_{G})\le cby Theorem 2.1, and hence χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=c.For the converse, since ω(GG)=χ(GG)=c\omega \left({{\mathcal{G}}}_{G})=\chi \left({{\mathcal{G}}}_{G})=c, the graph GG{{\mathcal{G}}}_{G}contains a cc-clique, which implies that GGcontains an abelian subgroup AAof order cc.□Corollary 2.3A finite group GGcontains an abelian subgroup AAof order ccif and only if κ(G)=Λ(G)=c\kappa \left(G)=\Lambda \left(G)=c.ProofIf GG{{\mathcal{G}}}_{G}is the commuting graph of GG, then κ(G)=ω(GG)\kappa \left(G)=\omega \left({{\mathcal{G}}}_{G}), Λ(G)=χ(GG)\Lambda \left(G)=\chi \left({{\mathcal{G}}}_{G})and Theorem 2.2 applies.□Proposition 2.4Let GGbe a finite group. Then the following statements are equivalent. (1)GGcontains a commuting set with ccelements.(2)GGcontains an abelian subgroup of order cc.(3)GG{{\mathcal{G}}}_{G}contains a cc-clique.(4)GG{{\mathcal{G}}}_{G}is a weakly perfect graph and χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=c.(5)GG{{\mathcal{G}}}_{G}contains a complete subgraph on ccvertices.(6)χ(GG)=κ(G)=c\chi \left({{\mathcal{G}}}_{G})=\kappa \left(G)=c.(7)ω(GG)=Λ(G)=c\omega \left({{\mathcal{G}}}_{G})=\Lambda \left(G)=c.Proof(1)⇒(2)Note that if XXis a commuting set of GG, then ⟨X⟩\langle X\rangle is a complete subgraph of GG{{\mathcal{G}}}_{G}on ccvertices. ⟨X⟩\langle X\rangle is in fact a cc-clique and we have already seen that this implies that XXis an abelian subgroup.(2)⇒(3)If GGcontains a commuting set with ccelements, GG{{\mathcal{G}}}_{G}contains a cc-clique. By Theorem 2.1, GGcontains an abelian subgroup of order cc.(3)⇒(4)If GG{{\mathcal{G}}}_{G}contains a cc-clique, GGcontains an abelian subgroup of order cc, by Corollary 2.2 GG{{\mathcal{G}}}_{G}is weakly perfect and χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=c.(4)⇒(5)GG{{\mathcal{G}}}_{G}weakly perfect implies ω(GG)=χ(GG)\omega \left({{\mathcal{G}}}_{G})=\chi \left({{\mathcal{G}}}_{G}), thus there is a cc-clique.(5)⇒(6)GG{{\mathcal{G}}}_{G}can be cc-colored and does not contain a complete subgraph in more than ccvertices and therefore GGdoes not contain a commutative set with more than ccvertices. From (5), GG{{\mathcal{G}}}_{G}cannot be kk-colored for k<ck\lt cand GGcontains a commutative set with ccelements. It follows at once that χ(GG)=κ(G)=c\chi \left({{\mathcal{G}}}_{G})=\kappa \left(G)=c.(6)⇒(7)Suppose χ(GG)=κ(G)=c\chi \left({{\mathcal{G}}}_{G})=\kappa \left(G)=c, since κ(G)=ω(GG)\kappa \left(G)=\omega \left({{\mathcal{G}}}_{G})and Λ(G)=χ(GG)\Lambda \left(G)=\chi \left({{\mathcal{G}}}_{G})we get the result.(7)⇒(1)ω(GG)=c\omega \left({{\mathcal{G}}}_{G})=cimplies the existence of a cc-clique, thus, there is commuting set of cardinality cc.□The following lemma given in [28] together with the fact that the degree of a vertex x∈GGx\in {{\mathcal{G}}}_{G}equals the order of its centralizer in GGminus 1 lead to Proposition 2.6.Lemma 2.5Let G{\mathcal{G}}be a graph with chromatic number χ(G)=q+1\chi \left({\mathcal{G}})=q+1and without any (q+1)\left(q+1)-cliques. Let T={x∈G:d(x)>q}T=\left\{x\in {\mathcal{G}}:d\left(x)\gt q\right\}, then(2)∑x∈T(d(x)−q)≥q−2.\sum _{x\in T}\left(d\left(x)-q)\ge q-2.Proposition 2.6Let GGbe a finite group. If χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=cand the set S={x∈G:∣CG(x)∣>c}S=\left\{x\in G:| {C}_{G}\left(x)| \gt c\right\}satisfies(3)∑x∈S(∣CG(x)∣−c)<c−3,\sum _{x\in S}\left(| {C}_{G}\left(x)| -c)\lt c-3,then GGcontains an abelian subgroup of order cc.ProofIf GGdoes not contain such an abelian subgroup, Theorem 2.1 says that GG{{\mathcal{G}}}_{G}does not contain any cc-clique, hence we may apply Lemma 2.5 with q=c−1q=c-1, if T={x∈V(GG):d(x)>c−1}T=\left\{x\in V\left({{\mathcal{G}}}_{G}):d\left(x)\gt c-1\right\}, then (4)∑x∈T(d(x)−c+1)≥c−3.\sum _{x\in T}\left(d\left(x)-c+1)\ge c-3.But d(x)=∣CG(x)∣−1d\left(x)=| {C}_{G}\left(x)| -1for each xx. Hence, S=TS=Tand inequality (4) becomes ∑x∈S(∣CG(x)∣−c)≥c−3.□\hspace{17.75em}\sum _{x\in S}\left(| {C}_{G}\left(x)| -c)\ge c-3.\hspace{17em}\square Proposition 2.7Let GGbe a finite group. If GGsatisfies the hypothesis of Proposition 2.6, then GGis abelian of order >3\gt 3.ProofIf S=∅S=\varnothing , then left hand side of inequality (3) is 0 and therefore we must have c>3c\gt 3. Moreover, note that ∣G∣=∣CG(1)∣≤c=χ(G)≤∣V(G)∣| G| =| {C}_{G}\left(1)| \le c=\chi \left({\mathcal{G}})\le | V\left({\mathcal{G}})| and GGis abelian (otherwise some pair of vertices of GG{{\mathcal{G}}}_{G}could be colored with the same color).If SSis non-empty, then 1∈G1\in Gmust be in SSand hence ∣CG(1)∣−c=∣G∣−c<c−3| {C}_{G}\left(1)| -c=| G| -c\lt c-3. This implies that ∣G∣<2c−3| G| \lt 2c-3. On the other hand, since GGcontains an abelian subgroup of order cc, we have ∣G∣=ck| G| =ckfor some integer kkand c(k−2)<−3c\left(k-2)\lt -3. But this inequality holds just for k=1k=1and c>3c\gt 3, in particular we must have k=1k=1and hence GGis abelian of order c>3c\gt 3.□From Propositions 2.6 and 2.7, we obtain the following result.Theorem 2.8Let GGbe a finite group of order >3\gt 3with Λ(G)=c\Lambda \left(G)=cand consider S={x∈G:∣CG(x)∣>c}S=\left\{x\in G:| {C}_{G}\left(x)| \gt c\right\}. Then GGis abelian if and only if(5)∑x∈S(∣CG(x)∣−c)<c−3.\sum _{x\in S}\left(| {C}_{G}\left(x)| -c)\lt c-3.ProofIf GGis abelian, then ∣CG(x)∣=∣G∣=Λ(G)=c| {C}_{G}\left(x)| =| G| =\Lambda \left(G)=cfor all x∈Gx\in G, hence the left hand side in inequality (3) is 0, while the right hand side is ∣G∣−3| G| -3, a number greater than 0.For the converse, suppose that GGis a non-abelian group of order >3\gt 3. According to Proposition 2.7, the hypothesis of Proposition 2.6 is not satisfied by GG, so we must have χ(GG)≠c\chi \left({{\mathcal{G}}}_{G})\ne cor ∑x∈S(∣CG(x)∣−c)≥c−3.\sum _{x\in S}\left(| {C}_{G}\left(x)| -c)\ge c-3.Since χ(GG)=Λ(G)=c\chi \left({{\mathcal{G}}}_{G})=\Lambda \left(G)=c, the result follows.□2.2Groups for which the bound is attainedThe following results show a relation between the expression of the order of finite groups and the existence of abelian subgroups of order cc. In order to prove Theorems 2.9, 2.10 and 2.13 we use some known results from group theory.Theorem 2.9Let ppbe a prime and GGa group of order p3{p}^{3}, then GGhas an abelian subgroup of order cc.ProofIf GGis abelian, we are done. If not, it is known that ∣Z(G)∣=p| Z\left(G)| =pand the number NNof conjugacy classes is N=p+1p3∑∣CG(x)∣,N=p+\frac{1}{{p}^{3}}\sum | {C}_{G}\left(x)| ,where the sum ∑∣CG(x)∣\sum | {C}_{G}\left(x)| is taken over the non-central elements.Note that ∣CG(x)∣=p2| {C}_{G}\left(x)| ={p}^{2}for each non-central element, otherwise there exists some non-central elements such that ∣CG(x)∣=p| {C}_{G}\left(x)| =p. But Z(G)≤CG(x)Z\left(G)\le {C}_{G}\left(x)also has order pp; hence, Z(G)=CG(x)Z\left(G)={C}_{G}\left(x). Since g∈CG(x)g\in {C}_{G}\left(x), x∈Z(G)x\in Z\left(G)and this is a contradiction. Therefore, the number of conjugacy classes is N=p+(p2−1)N=p+\left({p}^{2}-1); ppwith just one element and (p2−1)\left({p}^{2}-1)with ppelements.Finally, if m=2p−1m=2p-1we have c=1+1+⋯+1+p+⋯+p=p2c=1+1+\cdots +1+p+\cdots +p={p}^{2}. Note that GGcontains a subgroup of order p2{p}^{2}, which is necessarily an abelian subgroup.□Theorem 2.10If ppis a prime number and PPis a non-abelian group of order p3{p}^{3}then, P×ZpP\times {{\mathbb{Z}}}_{p}contains an abelian subgroup of order cc.ProofFirst note that Z(P×Zp)=Z(P)×Z(Zp)=Z(P)×ZpZ\left(P\times {{\mathbb{Z}}}_{p})=Z\left(P)\times Z\left({{\mathbb{Z}}}_{p})=Z\left(P)\times {{\mathbb{Z}}}_{p}. Now, since PPis non-abelian, ∣Z(P)∣=p| Z\left(P)| =pand therefore ∣Z(P×Zp)∣=p2| Z\left(P\times {{\mathbb{Z}}}_{p})| ={p}^{2}. Also, we have CP×Zp(x,y)=CP(x)×CZp(y)=CP(x)×Zp{C}_{P\times {{\mathbb{Z}}}_{p}}\left(x,y)={C}_{P}\left(x)\times {C}_{{{\mathbb{Z}}}_{p}}(y)={C}_{P}\left(x)\times {{\mathbb{Z}}}_{p}for each (x,y)∈P×Zp\left(x,y)\in P\times {{\mathbb{Z}}}_{p}. If x∈Px\in Pis a non-central element, we have already seen in the proof of Theorem 2.9 that ∣CP(x)∣=p2| {C}_{P}\left(x)| ={p}^{2}, in particular, CP(x){C}_{P}\left(x)is an abelian subgroup of PP. Hence, if (x,y)∈P×Zp\left(x,y)\in P\times {{\mathbb{Z}}}_{p}is a non-central element, then CP×Zp(x,y){C}_{P\times {{\mathbb{Z}}}_{p}}\left(x,y)is an abelian subgroup of order p3{p}^{3}.According to Burnside’s lemma, there are p2{p}^{2}conjugacy classes of order 1 and p3−p{p}^{3}-pconjugacy classes of order pp. Hence, m=2p2−pm=2{p}^{2}-pin Theorem 1.2 and consequently c=p2+(p2−p)p=p3c={p}^{2}+\left({p}^{2}-p)p={p}^{3}. If x∈P−Z(P)x\in P-Z\left(P)and y∈Zpy\in {{\mathbb{Z}}}_{p}is any element, then (x,y)\left(x,y)is a non-central element of P×ZpP\times {{\mathbb{Z}}}_{p}and CP×Zp(x,y){C}_{P\times {{\mathbb{Z}}}_{p}}\left(x,y)is an abelian subgroup of order cc.□Next we analyze special cases when the order is pnq{p}^{n}q. The following lemmas (see [30] and [31]) will be useful.Lemma 2.11If GGis a group of order pnq{p}^{n}q, where p>qp\gt qare primes, then GGcontains a unique Sylow pp-subgroup PP.Lemma 2.12If GGis a group which is not abelian, then G/Z(G)G\hspace{0.1em}\text{/}\hspace{0.1em}Z\left(G)is not cyclic. In particular, [G:Z(G)]\left[G:Z\left(G)]can never be a prime number.Theorem 2.13Let GGbe a non-abelian group of order pnq{p}^{n}q, where ppand qqare prime numbers: (1)If p>qp\gt qand the unique Sylow pp-subgroup of GGis abelian, then c=pnc={p}^{n}and GGcontains an abelian subgroup of order cc. In particular, for n=1,2n=1,2the group GGcontains an abelian subgroup of order cc.(2)If p<qp\lt qand n=2n=2, then either c=4c=4, c=pqc=pqor c=qc=q. In all these cases GGcontains an abelian subgroup of order cc.Moreover, if p2{p}^{2}does not divide q−1q-1, the case c=qc=qis not possible.Proof(1) Let PPbe the unique Sylow pp-subgroup of GG, note that all elements of this Sylow pp-subgroup have the smallest conjugacy classes with one or qqelements and note also that actually there exist elements x∈Px\in Pwith ∣[x]∣=q| \left[x]| =q(otherwise Z(G)=GZ\left(G)=Gor ∣Z(G)∣=pn| Z\left(G)| ={p}^{n}contradicting hypothesis or Lemma 2.12 in any case).Finally, let [x1],[x2],…,[xk]\left[{x}_{1}],\left[{x}_{2}],\ldots ,\left[{x}_{k}]be the conjugacy classes of the elements in PPordered increasingly by size. Since PPis the union of these conjugacy classes we have pn=∣P∣=∑i=1k∣[xi]∣=∣CG(xk)∣=c.{p}^{n}=| P| =\mathop{\sum }\limits_{i=1}^{k}| \left[{x}_{i}]| =| {C}_{G}\left({x}_{k})| =c.If n=1n=1, 2 the unique Sylow pp-subgroup of GGis abelian.(2) First, we examine the possible numbers of Sylow qq-subgroups and Sylow pp-subgroups. Let nq{n}_{q}and np{n}_{p}be the numbers of Sylow qq-subgroups and Sylow pp-subgroups, respectively. Since q>pq\gt p, q∤(p−1)q\nmid \left(p-1), it follows that nq≠p{n}_{q}\ne p. Now, if nq=p2{n}_{q}={p}^{2}, then necessarily q∣(p+1)q| \left(p+1), and hence q=p+1q=p+1. But the only prime numbers satisfying this condition are p=2p=2and q=3q=3. Consequently, ∣G∣=12| G| =12and then G≅A4,D12,Z3⋊Z4G\cong {A}_{4},{D}_{12},{{\mathbb{Z}}}_{3}\hspace{0.25em}\rtimes \hspace{0.25em}{{\mathbb{Z}}}_{4}. In the first case c=4c=4and in the other cases c=6c=6. A glance to the subgroups of GGshows the existence of an abelian subgroup of order cc.The other possibility is nq=1{n}_{q}=1, let QQbe such a Sylow qq-subgroup. Let us now consider the possible values for np{n}_{p}. Note that such a number cannot be equal to 1, otherwise GGis the direct product of these Sylow subgroups, and being each abelian, GGmust be abelian. This yields to np=q{n}_{p}=q.For analyzing the order of Z(G)Z\left(G), Lemma 2.12 ensures that ∣Z(G)∣≠pq,p2| Z\left(G)| \ne pq,{p}^{2}. On the other hand, if Z(G)Z\left(G)has order qq, then this is the unique Sylow qq-subgroup, so all the elements of the different Sylow pp-subgroups, except for the identity, are non-central elements. Hence, their centralizers have order p2{p}^{2}. But ∣Z(G)∣| Z\left(G)| divides the order of such centralizers, which is impossible. The only remaining possible values for ∣Z(G)∣| Z\left(G)| are ppor 1. Suppose that ∣Z(G)∣=p| Z\left(G)| =p. Since each Sylow pp-subgroup is a centralizer, Z(G)Z\left(G)is contained in each Sylow pp-subgroup. Thus, the sum of ∣Z(G)∣| Z\left(G)| and the number of non-central elements in each Sylow (ppor qq)-subgroup is p+q(p2−p)+(q−1)=p2q−pq+p+q−1p+q\left({p}^{2}-p)+\left(q-1)={p}^{2}q-pq+p+q-1. Subtracting this sum from p2q{p}^{2}qwe get pq−p−q+1pq-p-q+1, which corresponds to the number of non-central elements of GGthat do not belong to any Sylow (ppor qq)-subgroup.Now, the non-central elements of Sylow pp-subgroups and Sylow qq-subgroups have centralizers of order p2{p}^{2}and pqpq, respectively. The remaining pq−p−q+1pq-p-q+1elements have centralizers of order pqpq. Applying Burnside’s lemma we get: N=p+1p2q((q−1)pq+(pq−p−q+1)pq+q(p2−p)p2)=p+(q−1)+p(p−1),N=p+\frac{1}{{p}^{2}q}\left(\left(q-1)pq+\left(pq-p-q+1)pq+q\left({p}^{2}-p){p}^{2})=p+\left(q-1)+p\left(p-1),where p,q−1p,q-1and p(p−1)p\left(p-1)are the number of conjugacy classes of cardinally 1, ppand qq, respectively. Thus, the sum of the cardinals of the first ppconjugacy classes and the following q−1q-1conjugacy classes whose cardinals are ppis c=pqc=pq. If xxis an element not belonging to any Sylow (ppor qq)-subgroup, then ∣⟨x⟩∣=pq| \langle x\rangle | =pq.Suppose ∣Z(G)∣=1| Z\left(G)| =1, for x∈Qx\in Q, x≠1x\ne 1, we may observe that Q≤CG(x)Q\le {C}_{G}\left(x), thus, ∣CG(x)∣=q| {C}_{G}\left(x)| =qor pqpq. If ∣CG(x)∣=pq| {C}_{G}\left(x)| =pqthere exists y∈CG(x)y\in {C}_{G}\left(x)of order ppand, therefore, it must belong to a Sylow pp-subgroup PP, which implies that ∣P∩Q∣≥2| P\cap Q| \ge 2, a contradiction. Hence, ∣CG(x)∣=q| {C}_{G}\left(x)| =qand Q=CG(x)Q={C}_{G}\left(x). Moreover, the intersection of all Sylow pp-subgroups is trivial, otherwise we may obtain a subgroup of order ppcontained in all of them and the centralizer of a non-trivial element of this subgroup would have order >p2\gt {p}^{2}(it commutes with every element of these Sylow pp-subgroups). Now, if we take the sum over the number of elements in all Sylow subgroups we obtain q(p2−1)+(q−1)+1=p2qq\left({p}^{2}-1)+\left(q-1)+1={p}^{2}q, which means that GGis the union of its Sylow subgroups. Finally, calculating NNwe obtain N=1+1p2q((q−1)q+q(p2−1)p2)=1+q−1p2+(p2−1).N=1+\frac{1}{{p}^{2}q}\left(\left(q-1)q+q\left({p}^{2}-1){p}^{2})=1+\frac{q-1}{{p}^{2}}+\left({p}^{2}-1).So GGhas one conjugacy class with a unique element, q−1p2\frac{q-1}{{p}^{2}}conjugacy classes each with p2{p}^{2}elements and p2−1{p}^{2}-1conjugacy classes each with qqelements. This yields c=qc=qand the order of the unique Sylow qq-subgroup attains the bound. Notice that this case is not possible if p2{p}^{2}does not divide q−1q-1.□3Comparative analysis and conclusionThe main goal of our research was to determine under what conditions a finite group contains an abelian subgroup of maximal order. This interest is motivated by [24], where Bertram, relying on the commuting graph of a finite group, establishes that the order of every abelian subgroup of a finite group has an upper bound cc. A question that immediately arises is “under what conditions equality is attained?.” Precisely, Bertram proposed it in his work without finding the solution, he only mentioned as an example the solvable groups with a number of conjugation classes less than or equal to 7, except for G=Sym(4)G={\rm{Sym}}\left(4)(the symmetric group on 4 letters), where the bound cccoincides with the order of a centralizer, in fact, the largest centralizer other than GG. In this paper, we give a solution to this problem, that is, we find necessary and sufficient conditions on a finite group GGto contain an abelian subgroup with order the bound cc. In addition, we find some results that relate structural properties of the commuting graph GG{{\mathcal{G}}}_{G}to those of the underlying group GG, as well as some families of non-abelian groups attaining the upper bound cc.Over the years, other works have studied, in a certain way, abelian subgroups of maximal order of a finite group. For example, in [25] and [26] some arithmetic properties on the maximum order of an abelian subgroup are established, as well as their relation to the order of the group. In [25], it is shown that for any upper bound kkfor the order of the abelian subgroups of a finite group GGoccurs that ∣G∣| G| divides to k!k\&#x0021;. If no number less than ccsatisfies this property we conclude that GGcontains an abelian subgroup of order cc. In [27], abelian subgroups of maximal order of finite irreducible Coxeter groups are classified. Based on the development of our research, from the commuting graph of such Coxeter groups, it could be known which of these maximal order subgroups attains the bound cc. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Note on structural properties of graphs

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de Gruyter
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© 2022 Luis D. Arreola-Bautista et al., published by De Gruyter
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10.1515/math-2021-0137
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Abstract

1IntroductionOver the years, graphs generated from a group or a semigroup have been extensively studied. For example, in 1964, Bosak [1] studied graphs induced by semigroups; in [2] the intersection graphs of non-trivial subgroups of abelian groups with finite order were studied by Zelinka; in [3,4,5] the directed graph defined over the elements of a group, the well-known Cayley digraph, was studied; in the books in [6,7,8], numerous valuable applications of this kind of graph are presented, hence the importance of researching them. Another best-known graph is the so-called directed power graph, whose definition was given by Kelarev and Quinn [9], which is defined in such a way that it is possible to apply it also to semigroups, and thus, it was in [10,11,12] that power graphs of semigroups were considered for the first time. These papers use only the short-term “power graph.” However, it makes sense for both directed and undirected power graphs. In addition, Kelarev and Quinn [11] defined an interesting class of directed graphs over semigroups, the semigroup divisibility graphs. There is also one more example, the well-known hyperbolic graphs, mentioned in the works [13,14,15], whose main objects of study were initially Cayley graphs associated with finite groups, nowadays these graphs have several applications in Physics and Geometry. For more information on structural properties associated with a graph, see [16,17,18].It is said that group theory started with Galois (1811–1832), who showed that the best way to understand polynomials is by relating them to certain groups of permutations of their roots. From then onward, group theory has become a useful tool for different fields of mathematics, such as combinatorics, geometry, logic, number theory, and topology.At the end of the nineteenth century, there were two main streams of group theory: on the one hand topological groups (especially Lie groups); on the other hand, finite groups. The latter since its beginnings in the nineteenth century has grown to become an extensive and diverse part of algebra, in particular, the theory of locally finite groups and the theory of nilpotent, solvable groups [19,20]. In the early 1980s, this development culminated in the classification of finite simple groups, impressively and convincingly demonstrating the strength of its methods and results. Thanks to those works, all finite groups that are constructible from these simple groups are now known.Graph theory is an increasing mathematical discipline containing deep and strong results of high applicability. Its rapid development in the last few decades is not only due to its status as the main structure on which currently applied mathematics (computer science, combinatorial optimization, and operations research) is supported, but also due to its increasing connections in the applied sciences.In this paper, we are concerned with the commuting graph GG{{\mathcal{G}}}_{G}of a group GG. It is defined as the graph with GGas the set of vertices and where two of them form an edge if and only if they commute. This graph has been studied from several perspectives, for example, in [21] the chromatic and clique numbers are obtained for the commuting graphs of the dihedral-type groups; in [22] it is proved that the commuting graph of a finite minimal non-solvable group has diameter ≥3\ge 3; in [23] the authors obtain the number of spanning trees of the commuting graph for some specific groups, as well as the classification of finite groups for which the power graph and the commuting graph coincide; and in [24] properties of this graph, as coloring and independence number, are used to prove results about finite groups.The existence of abelian subgroups of maximal order of a given group GGis a topic widely studied in different papers. For example, in [25] it was shown that if mmis the maximal order of an abelian subgroup of a finite group GG, then ∣G∣| G| divides m!m\&#x0021;and in [26] some results are presented on mm. Moreover, in [27] a classification of the abelian subgroups of maximal order of finite irreducible Coxeter groups is obtained, the geometry of these subgroups is studied and some applications of such classification to statistical physics are given.In [24] Bertram, by using the commuting graph GG{{\mathcal{G}}}_{G}, gave an upper bound for the order of an abelian subgroup of a finite group GG. In addition, he proposed to find necessary and sufficient conditions for the existence of an abelian subgroup whose order attains the bound ∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{m}]| . In this paper, we give a solution to this problem and present some related results.In [28] the following result appears.Lemma 1.1If, for some qqthe number of vertices of degree ≥q\ge qis ≤q\le q, then the graph G{\mathcal{G}}can be qq-colored.And it is used in [24] to prove the following theorem.Theorem 1.2Let GGbe a finite group with conjugacy classes indexed by cardinality: 1=∣[x1]∣≤∣[x2]∣≤⋯,1=| \left[{x}_{1}]| \le | \left[{x}_{2}]| \hspace{0.33em}\le \cdots \hspace{0.33em},and let CG(x){C}_{G}\left(x)denote the centralizer of xx. If mmis the smallest number iisatisfying∣[x1]∣+∣[x2]∣+⋯+∣[xi]∣≥∣CG(xi)∣,| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{i}]| \ge | {C}_{G}\left({x}_{i})| ,then each abelian subgroup AAof GGsatisfies(1)∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣≥∣A∣.| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{m}]| \ge | A| .The problem proposed in [24] is the following:Find necessary and sufficient conditions on GGin order that inequality (1) in Theorem 1.2 becomes an equality for some abelian subgroup A<GA\lt G.In this paper, we use the following notation and known facts. Let G{\mathcal{G}}be an undirected simple graph, whose vertex set is V={x1,x2,…,xn}V=\left\{{x}_{1},{x}_{2},\ldots ,{x}_{n}\right\}. The degree of a vertex xxis the number of edges incident with xxand it is denoted by d(x)d\left(x). We say that the vertices of G{\mathcal{G}}can be kk-colored when there exists a partition of VVinto kksubsets, with no two vertices in the same subset connected by an edge of G{\mathcal{G}}. The minimum number kkfor which G{\mathcal{G}}can be kk-colored is called the chromatic number of G{\mathcal{G}}and it is denoted by χ(G)\chi \left({\mathcal{G}}). A complete subgraph of G{\mathcal{G}}is a subgraph when every pair of vertices is connected by an edge. A maximal complete subgraph of G{\mathcal{G}}is called a clique; we also use the term kk-clique for a clique consisting of kkvertices. The clique number ω(G)\omega \left({\mathcal{G}})is the maximal size of a clique contained in G{\mathcal{G}}. Note that if G{\mathcal{G}}can be kk-colored, then the number of vertices in each clique (as well as in each complete subgraph) has cardinality ≤k\le k; hence, we always have ω(G)≤χ(G)\omega \left({\mathcal{G}})\le \chi \left({\mathcal{G}}). A graph G{\mathcal{G}}is called weakly perfect if ω(G)=χ(G)\omega \left({\mathcal{G}})=\chi \left({\mathcal{G}})(see [29]).A subset XXof a finite group GGis called a commuting set if xy=yxxy=yxfor any x,y∈Xx,y\in X. The commuting number κ(G)\kappa \left(G)of a finite group GGis the maximum cardinality of a commuting set. A subset Y⊆GY\subseteq Gis called an anti-commuting set if xy=yxxy=yximplies x=yx=yfor any x,y∈Yx,y\in Y. The Λ\Lambda -number of GGis the minimal number kksuch that GGcan be partitioned into kkanti-commuting subsets; it is denoted by Λ(G)\Lambda \left(G).From the proof of Theorem 1.2, we get the following remark.Remark 1.3The graph GG{{\mathcal{G}}}_{G}is ∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{m}]| -colorable, thus, χ(GG)≤∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣.\chi \left({{\mathcal{G}}}_{G})\le | \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +\hspace{0.33em}| \left[{x}_{m}]| .2ResultsThroughout this paper we denote by ccthe sum ∣[x1]∣+∣[x2]∣+⋯+∣[xm]∣| \left[{x}_{1}]| +| \left[{x}_{2}]| +\cdots +| \left[{x}_{m}]| , the left hand side of inequality (1). Note that ω(GG)≤χ(GG)≤c.\omega \left({{\mathcal{G}}}_{G})\le \chi \left({{\mathcal{G}}}_{G})\le c.2.1GG{{\mathcal{G}}}_{G}parameters and the bound ccTheorem 2.1A finite group GGcontains an abelian subgroup of order ccif and only if GG{{\mathcal{G}}}_{G}contains a cc-clique.Corollary 2.2A finite group GGcontains an abelian subgroup AAof order ccif and only if GG{{\mathcal{G}}}_{G}is a weakly perfect graph with χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=c.ProofIf GGcontains an abelian subgroup of order cc, then c≤ω(GG)≤χ(GG)≤cc\le \omega \left({{\mathcal{G}}}_{G})\le \chi \left({{\mathcal{G}}}_{G})\le cby Theorem 2.1, and hence χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=c.For the converse, since ω(GG)=χ(GG)=c\omega \left({{\mathcal{G}}}_{G})=\chi \left({{\mathcal{G}}}_{G})=c, the graph GG{{\mathcal{G}}}_{G}contains a cc-clique, which implies that GGcontains an abelian subgroup AAof order cc.□Corollary 2.3A finite group GGcontains an abelian subgroup AAof order ccif and only if κ(G)=Λ(G)=c\kappa \left(G)=\Lambda \left(G)=c.ProofIf GG{{\mathcal{G}}}_{G}is the commuting graph of GG, then κ(G)=ω(GG)\kappa \left(G)=\omega \left({{\mathcal{G}}}_{G}), Λ(G)=χ(GG)\Lambda \left(G)=\chi \left({{\mathcal{G}}}_{G})and Theorem 2.2 applies.□Proposition 2.4Let GGbe a finite group. Then the following statements are equivalent. (1)GGcontains a commuting set with ccelements.(2)GGcontains an abelian subgroup of order cc.(3)GG{{\mathcal{G}}}_{G}contains a cc-clique.(4)GG{{\mathcal{G}}}_{G}is a weakly perfect graph and χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=c.(5)GG{{\mathcal{G}}}_{G}contains a complete subgraph on ccvertices.(6)χ(GG)=κ(G)=c\chi \left({{\mathcal{G}}}_{G})=\kappa \left(G)=c.(7)ω(GG)=Λ(G)=c\omega \left({{\mathcal{G}}}_{G})=\Lambda \left(G)=c.Proof(1)⇒(2)Note that if XXis a commuting set of GG, then ⟨X⟩\langle X\rangle is a complete subgraph of GG{{\mathcal{G}}}_{G}on ccvertices. ⟨X⟩\langle X\rangle is in fact a cc-clique and we have already seen that this implies that XXis an abelian subgroup.(2)⇒(3)If GGcontains a commuting set with ccelements, GG{{\mathcal{G}}}_{G}contains a cc-clique. By Theorem 2.1, GGcontains an abelian subgroup of order cc.(3)⇒(4)If GG{{\mathcal{G}}}_{G}contains a cc-clique, GGcontains an abelian subgroup of order cc, by Corollary 2.2 GG{{\mathcal{G}}}_{G}is weakly perfect and χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=c.(4)⇒(5)GG{{\mathcal{G}}}_{G}weakly perfect implies ω(GG)=χ(GG)\omega \left({{\mathcal{G}}}_{G})=\chi \left({{\mathcal{G}}}_{G}), thus there is a cc-clique.(5)⇒(6)GG{{\mathcal{G}}}_{G}can be cc-colored and does not contain a complete subgraph in more than ccvertices and therefore GGdoes not contain a commutative set with more than ccvertices. From (5), GG{{\mathcal{G}}}_{G}cannot be kk-colored for k<ck\lt cand GGcontains a commutative set with ccelements. It follows at once that χ(GG)=κ(G)=c\chi \left({{\mathcal{G}}}_{G})=\kappa \left(G)=c.(6)⇒(7)Suppose χ(GG)=κ(G)=c\chi \left({{\mathcal{G}}}_{G})=\kappa \left(G)=c, since κ(G)=ω(GG)\kappa \left(G)=\omega \left({{\mathcal{G}}}_{G})and Λ(G)=χ(GG)\Lambda \left(G)=\chi \left({{\mathcal{G}}}_{G})we get the result.(7)⇒(1)ω(GG)=c\omega \left({{\mathcal{G}}}_{G})=cimplies the existence of a cc-clique, thus, there is commuting set of cardinality cc.□The following lemma given in [28] together with the fact that the degree of a vertex x∈GGx\in {{\mathcal{G}}}_{G}equals the order of its centralizer in GGminus 1 lead to Proposition 2.6.Lemma 2.5Let G{\mathcal{G}}be a graph with chromatic number χ(G)=q+1\chi \left({\mathcal{G}})=q+1and without any (q+1)\left(q+1)-cliques. Let T={x∈G:d(x)>q}T=\left\{x\in {\mathcal{G}}:d\left(x)\gt q\right\}, then(2)∑x∈T(d(x)−q)≥q−2.\sum _{x\in T}\left(d\left(x)-q)\ge q-2.Proposition 2.6Let GGbe a finite group. If χ(GG)=c\chi \left({{\mathcal{G}}}_{G})=cand the set S={x∈G:∣CG(x)∣>c}S=\left\{x\in G:| {C}_{G}\left(x)| \gt c\right\}satisfies(3)∑x∈S(∣CG(x)∣−c)<c−3,\sum _{x\in S}\left(| {C}_{G}\left(x)| -c)\lt c-3,then GGcontains an abelian subgroup of order cc.ProofIf GGdoes not contain such an abelian subgroup, Theorem 2.1 says that GG{{\mathcal{G}}}_{G}does not contain any cc-clique, hence we may apply Lemma 2.5 with q=c−1q=c-1, if T={x∈V(GG):d(x)>c−1}T=\left\{x\in V\left({{\mathcal{G}}}_{G}):d\left(x)\gt c-1\right\}, then (4)∑x∈T(d(x)−c+1)≥c−3.\sum _{x\in T}\left(d\left(x)-c+1)\ge c-3.But d(x)=∣CG(x)∣−1d\left(x)=| {C}_{G}\left(x)| -1for each xx. Hence, S=TS=Tand inequality (4) becomes ∑x∈S(∣CG(x)∣−c)≥c−3.□\hspace{17.75em}\sum _{x\in S}\left(| {C}_{G}\left(x)| -c)\ge c-3.\hspace{17em}\square Proposition 2.7Let GGbe a finite group. If GGsatisfies the hypothesis of Proposition 2.6, then GGis abelian of order >3\gt 3.ProofIf S=∅S=\varnothing , then left hand side of inequality (3) is 0 and therefore we must have c>3c\gt 3. Moreover, note that ∣G∣=∣CG(1)∣≤c=χ(G)≤∣V(G)∣| G| =| {C}_{G}\left(1)| \le c=\chi \left({\mathcal{G}})\le | V\left({\mathcal{G}})| and GGis abelian (otherwise some pair of vertices of GG{{\mathcal{G}}}_{G}could be colored with the same color).If SSis non-empty, then 1∈G1\in Gmust be in SSand hence ∣CG(1)∣−c=∣G∣−c<c−3| {C}_{G}\left(1)| -c=| G| -c\lt c-3. This implies that ∣G∣<2c−3| G| \lt 2c-3. On the other hand, since GGcontains an abelian subgroup of order cc, we have ∣G∣=ck| G| =ckfor some integer kkand c(k−2)<−3c\left(k-2)\lt -3. But this inequality holds just for k=1k=1and c>3c\gt 3, in particular we must have k=1k=1and hence GGis abelian of order c>3c\gt 3.□From Propositions 2.6 and 2.7, we obtain the following result.Theorem 2.8Let GGbe a finite group of order >3\gt 3with Λ(G)=c\Lambda \left(G)=cand consider S={x∈G:∣CG(x)∣>c}S=\left\{x\in G:| {C}_{G}\left(x)| \gt c\right\}. Then GGis abelian if and only if(5)∑x∈S(∣CG(x)∣−c)<c−3.\sum _{x\in S}\left(| {C}_{G}\left(x)| -c)\lt c-3.ProofIf GGis abelian, then ∣CG(x)∣=∣G∣=Λ(G)=c| {C}_{G}\left(x)| =| G| =\Lambda \left(G)=cfor all x∈Gx\in G, hence the left hand side in inequality (3) is 0, while the right hand side is ∣G∣−3| G| -3, a number greater than 0.For the converse, suppose that GGis a non-abelian group of order >3\gt 3. According to Proposition 2.7, the hypothesis of Proposition 2.6 is not satisfied by GG, so we must have χ(GG)≠c\chi \left({{\mathcal{G}}}_{G})\ne cor ∑x∈S(∣CG(x)∣−c)≥c−3.\sum _{x\in S}\left(| {C}_{G}\left(x)| -c)\ge c-3.Since χ(GG)=Λ(G)=c\chi \left({{\mathcal{G}}}_{G})=\Lambda \left(G)=c, the result follows.□2.2Groups for which the bound is attainedThe following results show a relation between the expression of the order of finite groups and the existence of abelian subgroups of order cc. In order to prove Theorems 2.9, 2.10 and 2.13 we use some known results from group theory.Theorem 2.9Let ppbe a prime and GGa group of order p3{p}^{3}, then GGhas an abelian subgroup of order cc.ProofIf GGis abelian, we are done. If not, it is known that ∣Z(G)∣=p| Z\left(G)| =pand the number NNof conjugacy classes is N=p+1p3∑∣CG(x)∣,N=p+\frac{1}{{p}^{3}}\sum | {C}_{G}\left(x)| ,where the sum ∑∣CG(x)∣\sum | {C}_{G}\left(x)| is taken over the non-central elements.Note that ∣CG(x)∣=p2| {C}_{G}\left(x)| ={p}^{2}for each non-central element, otherwise there exists some non-central elements such that ∣CG(x)∣=p| {C}_{G}\left(x)| =p. But Z(G)≤CG(x)Z\left(G)\le {C}_{G}\left(x)also has order pp; hence, Z(G)=CG(x)Z\left(G)={C}_{G}\left(x). Since g∈CG(x)g\in {C}_{G}\left(x), x∈Z(G)x\in Z\left(G)and this is a contradiction. Therefore, the number of conjugacy classes is N=p+(p2−1)N=p+\left({p}^{2}-1); ppwith just one element and (p2−1)\left({p}^{2}-1)with ppelements.Finally, if m=2p−1m=2p-1we have c=1+1+⋯+1+p+⋯+p=p2c=1+1+\cdots +1+p+\cdots +p={p}^{2}. Note that GGcontains a subgroup of order p2{p}^{2}, which is necessarily an abelian subgroup.□Theorem 2.10If ppis a prime number and PPis a non-abelian group of order p3{p}^{3}then, P×ZpP\times {{\mathbb{Z}}}_{p}contains an abelian subgroup of order cc.ProofFirst note that Z(P×Zp)=Z(P)×Z(Zp)=Z(P)×ZpZ\left(P\times {{\mathbb{Z}}}_{p})=Z\left(P)\times Z\left({{\mathbb{Z}}}_{p})=Z\left(P)\times {{\mathbb{Z}}}_{p}. Now, since PPis non-abelian, ∣Z(P)∣=p| Z\left(P)| =pand therefore ∣Z(P×Zp)∣=p2| Z\left(P\times {{\mathbb{Z}}}_{p})| ={p}^{2}. Also, we have CP×Zp(x,y)=CP(x)×CZp(y)=CP(x)×Zp{C}_{P\times {{\mathbb{Z}}}_{p}}\left(x,y)={C}_{P}\left(x)\times {C}_{{{\mathbb{Z}}}_{p}}(y)={C}_{P}\left(x)\times {{\mathbb{Z}}}_{p}for each (x,y)∈P×Zp\left(x,y)\in P\times {{\mathbb{Z}}}_{p}. If x∈Px\in Pis a non-central element, we have already seen in the proof of Theorem 2.9 that ∣CP(x)∣=p2| {C}_{P}\left(x)| ={p}^{2}, in particular, CP(x){C}_{P}\left(x)is an abelian subgroup of PP. Hence, if (x,y)∈P×Zp\left(x,y)\in P\times {{\mathbb{Z}}}_{p}is a non-central element, then CP×Zp(x,y){C}_{P\times {{\mathbb{Z}}}_{p}}\left(x,y)is an abelian subgroup of order p3{p}^{3}.According to Burnside’s lemma, there are p2{p}^{2}conjugacy classes of order 1 and p3−p{p}^{3}-pconjugacy classes of order pp. Hence, m=2p2−pm=2{p}^{2}-pin Theorem 1.2 and consequently c=p2+(p2−p)p=p3c={p}^{2}+\left({p}^{2}-p)p={p}^{3}. If x∈P−Z(P)x\in P-Z\left(P)and y∈Zpy\in {{\mathbb{Z}}}_{p}is any element, then (x,y)\left(x,y)is a non-central element of P×ZpP\times {{\mathbb{Z}}}_{p}and CP×Zp(x,y){C}_{P\times {{\mathbb{Z}}}_{p}}\left(x,y)is an abelian subgroup of order cc.□Next we analyze special cases when the order is pnq{p}^{n}q. The following lemmas (see [30] and [31]) will be useful.Lemma 2.11If GGis a group of order pnq{p}^{n}q, where p>qp\gt qare primes, then GGcontains a unique Sylow pp-subgroup PP.Lemma 2.12If GGis a group which is not abelian, then G/Z(G)G\hspace{0.1em}\text{/}\hspace{0.1em}Z\left(G)is not cyclic. In particular, [G:Z(G)]\left[G:Z\left(G)]can never be a prime number.Theorem 2.13Let GGbe a non-abelian group of order pnq{p}^{n}q, where ppand qqare prime numbers: (1)If p>qp\gt qand the unique Sylow pp-subgroup of GGis abelian, then c=pnc={p}^{n}and GGcontains an abelian subgroup of order cc. In particular, for n=1,2n=1,2the group GGcontains an abelian subgroup of order cc.(2)If p<qp\lt qand n=2n=2, then either c=4c=4, c=pqc=pqor c=qc=q. In all these cases GGcontains an abelian subgroup of order cc.Moreover, if p2{p}^{2}does not divide q−1q-1, the case c=qc=qis not possible.Proof(1) Let PPbe the unique Sylow pp-subgroup of GG, note that all elements of this Sylow pp-subgroup have the smallest conjugacy classes with one or qqelements and note also that actually there exist elements x∈Px\in Pwith ∣[x]∣=q| \left[x]| =q(otherwise Z(G)=GZ\left(G)=Gor ∣Z(G)∣=pn| Z\left(G)| ={p}^{n}contradicting hypothesis or Lemma 2.12 in any case).Finally, let [x1],[x2],…,[xk]\left[{x}_{1}],\left[{x}_{2}],\ldots ,\left[{x}_{k}]be the conjugacy classes of the elements in PPordered increasingly by size. Since PPis the union of these conjugacy classes we have pn=∣P∣=∑i=1k∣[xi]∣=∣CG(xk)∣=c.{p}^{n}=| P| =\mathop{\sum }\limits_{i=1}^{k}| \left[{x}_{i}]| =| {C}_{G}\left({x}_{k})| =c.If n=1n=1, 2 the unique Sylow pp-subgroup of GGis abelian.(2) First, we examine the possible numbers of Sylow qq-subgroups and Sylow pp-subgroups. Let nq{n}_{q}and np{n}_{p}be the numbers of Sylow qq-subgroups and Sylow pp-subgroups, respectively. Since q>pq\gt p, q∤(p−1)q\nmid \left(p-1), it follows that nq≠p{n}_{q}\ne p. Now, if nq=p2{n}_{q}={p}^{2}, then necessarily q∣(p+1)q| \left(p+1), and hence q=p+1q=p+1. But the only prime numbers satisfying this condition are p=2p=2and q=3q=3. Consequently, ∣G∣=12| G| =12and then G≅A4,D12,Z3⋊Z4G\cong {A}_{4},{D}_{12},{{\mathbb{Z}}}_{3}\hspace{0.25em}\rtimes \hspace{0.25em}{{\mathbb{Z}}}_{4}. In the first case c=4c=4and in the other cases c=6c=6. A glance to the subgroups of GGshows the existence of an abelian subgroup of order cc.The other possibility is nq=1{n}_{q}=1, let QQbe such a Sylow qq-subgroup. Let us now consider the possible values for np{n}_{p}. Note that such a number cannot be equal to 1, otherwise GGis the direct product of these Sylow subgroups, and being each abelian, GGmust be abelian. This yields to np=q{n}_{p}=q.For analyzing the order of Z(G)Z\left(G), Lemma 2.12 ensures that ∣Z(G)∣≠pq,p2| Z\left(G)| \ne pq,{p}^{2}. On the other hand, if Z(G)Z\left(G)has order qq, then this is the unique Sylow qq-subgroup, so all the elements of the different Sylow pp-subgroups, except for the identity, are non-central elements. Hence, their centralizers have order p2{p}^{2}. But ∣Z(G)∣| Z\left(G)| divides the order of such centralizers, which is impossible. The only remaining possible values for ∣Z(G)∣| Z\left(G)| are ppor 1. Suppose that ∣Z(G)∣=p| Z\left(G)| =p. Since each Sylow pp-subgroup is a centralizer, Z(G)Z\left(G)is contained in each Sylow pp-subgroup. Thus, the sum of ∣Z(G)∣| Z\left(G)| and the number of non-central elements in each Sylow (ppor qq)-subgroup is p+q(p2−p)+(q−1)=p2q−pq+p+q−1p+q\left({p}^{2}-p)+\left(q-1)={p}^{2}q-pq+p+q-1. Subtracting this sum from p2q{p}^{2}qwe get pq−p−q+1pq-p-q+1, which corresponds to the number of non-central elements of GGthat do not belong to any Sylow (ppor qq)-subgroup.Now, the non-central elements of Sylow pp-subgroups and Sylow qq-subgroups have centralizers of order p2{p}^{2}and pqpq, respectively. The remaining pq−p−q+1pq-p-q+1elements have centralizers of order pqpq. Applying Burnside’s lemma we get: N=p+1p2q((q−1)pq+(pq−p−q+1)pq+q(p2−p)p2)=p+(q−1)+p(p−1),N=p+\frac{1}{{p}^{2}q}\left(\left(q-1)pq+\left(pq-p-q+1)pq+q\left({p}^{2}-p){p}^{2})=p+\left(q-1)+p\left(p-1),where p,q−1p,q-1and p(p−1)p\left(p-1)are the number of conjugacy classes of cardinally 1, ppand qq, respectively. Thus, the sum of the cardinals of the first ppconjugacy classes and the following q−1q-1conjugacy classes whose cardinals are ppis c=pqc=pq. If xxis an element not belonging to any Sylow (ppor qq)-subgroup, then ∣⟨x⟩∣=pq| \langle x\rangle | =pq.Suppose ∣Z(G)∣=1| Z\left(G)| =1, for x∈Qx\in Q, x≠1x\ne 1, we may observe that Q≤CG(x)Q\le {C}_{G}\left(x), thus, ∣CG(x)∣=q| {C}_{G}\left(x)| =qor pqpq. If ∣CG(x)∣=pq| {C}_{G}\left(x)| =pqthere exists y∈CG(x)y\in {C}_{G}\left(x)of order ppand, therefore, it must belong to a Sylow pp-subgroup PP, which implies that ∣P∩Q∣≥2| P\cap Q| \ge 2, a contradiction. Hence, ∣CG(x)∣=q| {C}_{G}\left(x)| =qand Q=CG(x)Q={C}_{G}\left(x). Moreover, the intersection of all Sylow pp-subgroups is trivial, otherwise we may obtain a subgroup of order ppcontained in all of them and the centralizer of a non-trivial element of this subgroup would have order >p2\gt {p}^{2}(it commutes with every element of these Sylow pp-subgroups). Now, if we take the sum over the number of elements in all Sylow subgroups we obtain q(p2−1)+(q−1)+1=p2qq\left({p}^{2}-1)+\left(q-1)+1={p}^{2}q, which means that GGis the union of its Sylow subgroups. Finally, calculating NNwe obtain N=1+1p2q((q−1)q+q(p2−1)p2)=1+q−1p2+(p2−1).N=1+\frac{1}{{p}^{2}q}\left(\left(q-1)q+q\left({p}^{2}-1){p}^{2})=1+\frac{q-1}{{p}^{2}}+\left({p}^{2}-1).So GGhas one conjugacy class with a unique element, q−1p2\frac{q-1}{{p}^{2}}conjugacy classes each with p2{p}^{2}elements and p2−1{p}^{2}-1conjugacy classes each with qqelements. This yields c=qc=qand the order of the unique Sylow qq-subgroup attains the bound. Notice that this case is not possible if p2{p}^{2}does not divide q−1q-1.□3Comparative analysis and conclusionThe main goal of our research was to determine under what conditions a finite group contains an abelian subgroup of maximal order. This interest is motivated by [24], where Bertram, relying on the commuting graph of a finite group, establishes that the order of every abelian subgroup of a finite group has an upper bound cc. A question that immediately arises is “under what conditions equality is attained?.” Precisely, Bertram proposed it in his work without finding the solution, he only mentioned as an example the solvable groups with a number of conjugation classes less than or equal to 7, except for G=Sym(4)G={\rm{Sym}}\left(4)(the symmetric group on 4 letters), where the bound cccoincides with the order of a centralizer, in fact, the largest centralizer other than GG. In this paper, we give a solution to this problem, that is, we find necessary and sufficient conditions on a finite group GGto contain an abelian subgroup with order the bound cc. In addition, we find some results that relate structural properties of the commuting graph GG{{\mathcal{G}}}_{G}to those of the underlying group GG, as well as some families of non-abelian groups attaining the upper bound cc.Over the years, other works have studied, in a certain way, abelian subgroups of maximal order of a finite group. For example, in [25] and [26] some arithmetic properties on the maximum order of an abelian subgroup are established, as well as their relation to the order of the group. In [25], it is shown that for any upper bound kkfor the order of the abelian subgroups of a finite group GGoccurs that ∣G∣| G| divides to k!k\&#x0021;. If no number less than ccsatisfies this property we conclude that GGcontains an abelian subgroup of order cc. In [27], abelian subgroups of maximal order of finite irreducible Coxeter groups are classified. Based on the development of our research, from the commuting graph of such Coxeter groups, it could be known which of these maximal order subgroups attains the bound cc.

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: Abelian subgroup; commuting graph; clique; chromatic number; weakly perfect graph; 05C75

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