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Publicly Available Published by De Gruyter March 10, 2020

Groups in which the centralizer of any non-central element is maximal

  • Xianhe Zhao EMAIL logo , Ruifang Chen and Xiuyun Guo
From the journal Journal of Group Theory

Abstract

Let x be an element of a finite group G. It is clear that xCG(x)G. For the cases where CG(x)=G and CG(x)=x for any element x of G, the structure of G is easily decided. In this paper, we investigate the case where CG(x) is maximal in G and obtain the structure of G.

1 Introduction

The relationship between the centralizers of elements and the structure of groups has been studied extensively. In the mid 20th century, the authors of [4, 10, 11] studied the finite groups in which centralizers of some elements are nilpotent, even abelian. Later on, the author of [7] investigated the relationship between the solvability of finite groups and the generalized normality of the centralizers of some elements. Recently, many authors studied the influence of the number of centralizers on the structure of groups. For example, see [6, 1, 12] for finite groups and [13] for infinite groups.

Now let G be a finite group, Z(G) the center of G, and x an element of G. We write “¯” for the factor group modulo Z(G), set xG for the conjugacy class of x in G and |xG| for the size of xG. Apart from this, we employ HG to denote that H is a maximal subgroup of G.

The inclusion relationship xCG(x)G is always true if x is an element of a finite group G. It easily follows that G is abelian if and only if CG(x)=G for any x in G and that |G| is square free if CG(x)=x for any element x in G. Naturally, we may ask:

Question.

What is the structure of a finite group G if CG(x)G for any non-central element x of G?

From now on, whenever centralizer is mentioned, it means centralizer of a non-central element. We say that a group G is an 𝒞-group (or G𝒞) if all of its centralizers are maximal in G. Recall that a group G is said to be inner abelian if G is not abelian but every proper subgroup of G is abelian.

It is known that, for a finite 𝒞-group G, its structure depends on whether |G¯| is a prime power or not. So our main results are divided into the following two theorems.

Theorem A.

Let G be a finite non-abelian group such that |G¯| is not a prime power. If GMC, then either G¯ is abelian or G¯=P¯Q¯ is an inner abelian group with order |P¯|=pa and |Q¯|=q, where p and q are two different primes, and a is a positive integer.

Theorem B.

Let G be a finite non-abelian group such that G¯ is a p-group. If GMC, then G¯ is an elementary abelian p-group.

2 Preliminaries

In this section, we first state some lemmas which are needed for the proof of our main result.

Lemma 2.1 ([2, Lemma 1.1]).

Let N be a normal subgroup of a finite group G and x an element of G. Then

  1. |xN| divides |xG|;

  2. |(Nx)G/N| divides |xG|.

Lemma 2.2 ([5, Lemma 2.3]).

Let π be a set of some primes and N a normal subgroup of a finite group G. If xN is a π-element, then there exists a π-element x* of G such that xN=x*N.

Lemma 2.3 ([3, Theorem 1]).

Let G be a finite group acting transitively on a set Ω with |Ω|>1. Then there exists a prime p and a p-element x of G such that x acts without fixed points on Ω.

Lemma 2.4 ([8, Theorem 3.2]).

Let G be a finite group and p a prime, p3,5. Then G has abelian Sylow p-subgroups if and only if p|xG| for all p-elements x of G.

Lemma 2.5 ([9, Theorem 10.1.8]).

Let p be a prime factor of the order of a finite group G, P a Sylow p-subgroup of G. If NG(P)=CG(P), then G is p-nilpotent.

Lemma 2.6.

Let G be a finite p-solvable group with p a prime factor of the order of G. If p|xG| for every p-element x of G, then the Sylow p-subgroups of G is abelian.

Proof.

The p-solvability of G implies Op(G)>1 or Op(G)>1. If Op(G)>1, then the hypotheses of the lemma for the quotient group G/Op(G) are inherited by Lemma 2.1 (b). If P is a Sylow p-subgroup of G, then POp(G)/Op(G) is abelian by induction, and therefore P is abelian. So we may assume Op(G)>1. Let P be a Sylow p-subgroup. For any p-element x of P, we see Op(G)CG(x) by p|xG|, and therefore xCG(Op(G)). It follows that PCG(Op(G)). On the other hand, CG(Op(G))Op(G) by [9, Theorem 9.3.1]. Hence we have that P=Z(P)=Op(G) is abelian. ∎

3 Main results

In this section, we prove our main results. Let us start with the following lemma.

Lemma 3.1.

Let G be a finite group and x,yGZ(G) such that xy=yx and (o(x),o(y))=1. Then

  1. CG(x)CG(y)=CG(xy);

  2. if G𝒞, then CG(x)=CG(y)=CG(xy).

Proof.

(1) It is clear that CG(xy)CG((xy)o(x)), and so CG(xy)CG(yo(x)) by xy=yx. Since (o(x),o(y))=1, we see CG(yo(x))=CG(y). Thus we have CG(xy)CG(y). By similar arguments, we have CG(xy)CG(x), and hence CG(xy)CG(x)CG(y). On the other hand, it is clear that

CG(x)CG(y)CG(xy),

and therefore CG(x)CG(y)=CG(xy).

(2) Noting that xyGZ(G), we see that CG(xy) is also a maximal subgroup. It follows from (1) that CG(xy)=CG(x). Similarly, CG(xy)=CG(y), so CG(x)=CG(y)=CG(xy). ∎

Lemma 3.2.

Let G be a finite non-abelian group, p a prime factor of |G|, and x a non-central element in G. If GMC and |G¯| is not a prime power, then

  1. CG(x)=Px×Lx, where Px is a Sylow p-subgroup and Lx is a Hall p-subgroup of CG(x). Especially, if Lx is not contained in Z(G), then CG(x) is abelian.

  2. Lx must be contained in Z(G).

Proof.

Let q (p) be a prime factor of |CG(x)| (otherwise, CG(x) is a p-group and there is nothing to be proved), and let y be a q-element of CG(x). Part (2) of Lemma 3.1 shows that CG(x)=CG(y), which forces us to deduce yZ(CG(x)). It follows that there exist a Sylow p-subgroup Px and a Hall p-subgroup Lx of CG(x) such that CG(x)=Px×Lx.

Especially, if LxZ(G), then there exists a non-central p-element zLx. Applying Lemma 3.1 (2) once again, we have CG(x)=CG(z), and therefore PxZ(CG(z)). It follows that PxZ(CG(x)), and so CG(x) is abelian. The proof of (I) is complete.

Now we prove (II). Suppose that Lx is not contained in Z(G). The proof can be divided into the following steps.

Step 1. It is clear that CG(x)¯ is not a prime power order group.

Step 2. There exists a prime power order non-central element y such that

CG(x)g¯CG(y)¯=1for everygG.

Since |xG|>1, there exists a prime power order element y acting on xG without fixed points by Lemma 2.3, which shows that xGCG(y)=. Note that CG(x) is abelian, so we have CG(x)g¯CG(y)¯=1 for any gG. Otherwise, there exists a prime power order element 1w¯CG(x)g¯CG(y)¯, where we may assume that w is a prime power order element by Lemma 2.2, which leads to CG(x)gCG(w) since CG(x) is abelian. Now the maximality of CG(x)g enters our argument, which shows CG(x)g=CG(w). While yCG(w), it follows that yCG(x)g, so xgCG(y), in contradiction to the fact xGCG(y)=.

Keeping in mind that CG(x)G, we have

CG(x)GorNG(CG(x))=CG(x).

So we would distinguish whether CG(x) is normal in G or not.

Case 1. CG(x)G. In this case, the proof is broken up into the following Step 3.2 to Step 3.2, each one in itself being fairly straightforward.

Step 3. G=CG(x)y and CG(y)¯=y¯=y¯ for y subject to the properties in Step 3.2. In fact, take an element y subject to the properties in Step 3.2. Since yCG(x) and CG(x) is maximal, we have G=CG(x)y. Thus

CG(y)=CG(y)G=CG(y)CG(x)y=(CG(y)CG(x))y.

Noticing that CG(y)CG(x)=Z(G) by Step 3.2, we see CG(y)¯=y¯=y¯.

Step 4. NG¯(y¯)=y¯. Otherwise, y¯G¯ by the maximality of CG(y). Since CG(x)¯y¯=CG(x)¯CG(y)¯=1 by Step 3.2, we see

(3.1)G¯=CG(x)¯×y¯.

On the other hand, keeping in mind that CG(y)¯=y¯, the maximality of CG(y) implies

(3.2)G¯=x¯×y¯.

Comparing equations (3.1) and (3.2), we can conclude that CG(x)¯=x¯, which shows that CG(x)¯ is a prime power order group, in contradiction to the Step 3.2.

Step 5. y¯y¯g¯=1 for every g¯G¯-y¯. For every g¯G¯-y¯, it is clear that y¯g¯y¯ by Step 3.2. Since y¯g¯=y¯g¯=yg¯=yg¯, we see yg¯y¯, which shows that ygy. If there exists an element 1t¯y¯y¯g¯, then t¯y¯yg¯=yyg¯. By Lemma 2.2, we may assume that tyyg. Thus yZ(G)<y,ygZ(G)CG(t). It follows that tZ(G) by the maximality of CG(y), a contradiction.

Step 6. The contradiction. In fact, by Step 3.2 and Step 3.2, G¯=CG(x)¯y¯ is a Frobenius group with abelian F-kernel CG(x)¯ and a prime power order F-complement y¯. Let P¯Sylp(CG(x)¯). Since CG(x)¯G¯ and CG(x)¯ is abelian, we see P¯G¯. It follows from 1P¯ and the maximality of y¯ that P¯y¯=G¯. Hence CG(x)¯=P¯, in contradiction to Step 3.2.

Case 2. NG(CG(x))=CG(x). For this case, the contradiction will be followed by the following two paragraphs.

Obviously, NG¯(CG(x)¯)=CG(x)¯. By using a similar argument to Step 3.2, we have CG(x)¯CG(x)¯g¯=1 for every g¯G¯-CG(x)¯, so G¯ is a Frobenius group with a F-complement CG(x)¯ and the F-kernel N¯.

Let 1Q¯Sylq(N¯). In view of the nilpotence of N¯ and the maximality of CG(x)¯, we may assume that N¯=Q¯, which shows that G¯=Q¯CG(x)¯. Taking 1z¯Z(Q¯), it is clear that z is a non-central q-element by Lemma 2.2. Noticing that the F-group G¯=Q¯CG(x)¯ and z¯Z(Q¯), we have CG¯(z¯)=Q¯. On the other hand, since CG(z)¯CG¯(z¯) and CG(z)¯G¯, we see CG(z)¯=CG¯(z¯)=Q¯. It follows that Q¯G¯ by the hypotheses of the theorem. Hence G¯=Q¯x¯, which shows that CG(x)¯=x¯, in contradiction to Step 3.2. ∎

Now, we are ready to prove Theorem A.

Theorem A.

Let G be a finite non-abelian group such that |G¯| is not a prime power. If GMC, then either G¯ is abelian or G¯=P¯Q¯ is an inner abelian group with order |P¯|=pa and |Q¯|=q, where p and q are two different primes, and a is a positive integer.

Proof.

Assume that G¯ is not abelian. The proof of the theorem will be accomplished step by step.

Step 1. Let p be a prime factor of |G| and x a non-central p-element of G. We have that CG(x)¯ is a Sylow p-subgroup of G¯, and therefore p|xG|. Let p be a prime factor of |G| and x a non-central p-element of G. Lemma 3.2 shows that CG(x)=Px×Lx and LxZ(G), so CG(x)¯ is a p-group. Keeping in mind that |G¯| is not a prime power, we see that CG(x)¯ is a Sylow p-subgroup of G¯ by the maximality of CG(x)¯, and hence p|xG|.

Step 2. G is solvable, and every Sylow subgroup of G is abelian. Let p be a prime factor of |G| and x a non-central p-element of G. Notice that Step A ensures that p|xG|, and we have that P is abelian when p{3,5} by Lemma 2.4. If 2|G¯|, then the odd order theorem implies that G¯ is solvable, and so is G. Assume that 2|G¯| and S is a Sylow 2-subgroup of G. It is clear that S is abelian, and so is S¯, which leads to S¯CG¯(S¯)NG¯(S¯). Also, the abelianess of S and Step A imply S¯G¯, which forces us to conclude that either S¯G¯ or NG¯(S¯)=CG¯(S¯)=S¯. Clearly, the former shows that G¯ is 2-closed, and the latter implies G¯ is 2-nilpotent by Lemma 2.5, all of which indicate that G¯ is solvable, and so is G.

For every prime factor p of |G| and every non-central p-element x of G, Step A admits that p|xG|. Since G is solvable, we see every Sylow p-subgroup of G is abelian by Lemma 2.6.

Step 3. G is either p-closed or p-nilpotent for every prime p|G¯|. Let p be an arbitrary prime factor of |G¯| and P a Sylow p-subgroup of G. Clearly, PZ(G). Also, P is abelian by Step A; it follows that P¯CG¯(P¯)NG¯(P¯). The maximality of P¯ in G¯ forces us to conclude that either P¯G¯ or NG¯(P¯)=CG¯(P¯)=P¯. Therefore, we can conclude that G is either p-closed or p-nilpotent.

Step 4. G is not nilpotent, and there at least exists a non-central Sylow p-subgroup P of G such that PG. If G is nilpotent, then, by Step A, we have that G is abelian, so is G¯, a contradiction. This indicates that G is not nilpotent. By Step A, it is clear that there exists at least a prime p dividing |G¯| such that PG, where P is a Sylow p-subgroup of G.

Step 5. G¯=P¯Q¯ is an inner abelian group with |Q¯|=q. By Step A, we may assume that P is a normal Sylow p-subgroup of G. Also, Step A implies that P¯G¯, so G¯=P¯Q¯. Taking y¯Q¯ with o(y¯)=q, we have G¯=P¯y¯ by the maximality of P¯ once again, so |Q¯|=q. Of course, G¯ is inner abelian by (II) of Lemma 3.2. ∎

Remark 1.

In Theorem A, if we set Z(G)=1, we have the following.

Corollary.

Let G be a finite group with Z(G)=1, and |G| not a prime power. If GMC, then G=PQ is an inner abelian group with order |P|=pa and |Q|=q, where p and q are two different primes, and a is a positive integer.

Theorem B.

Let G be a finite non-abelian group such that G¯ is a p-group. If GMC, then G¯ is an elementary abelian p-group.

Proof.

Let x be a non-central element of G, and set Gp=gpgG. For any tG, if tCG(x), then tpCG(x). If tCG(x), the maximality of CG(x) (note that, in this case, as NG(CG(x))CG(x), so CG(x) is normal in G) leads to G=CG(x)t. By computing the order of G, we have CG(x)tt, which yields CG(x)t=tp, so tpCG(x). Judging from the above two cases, we have GpCG(x). On the other hand, it is clear that CG(x)¯G¯ by G𝒞. Keeping in mind that G¯ is a p-group, we see that G¯/CG(x)¯ is abelian, and so is G/CG(x). This indicates that GCG(x). By the arbitrariness of x, we have that Φ(G)=GGp is contained in Z(G); it follows that G¯=G/Z(G) is an elementary abelian p-group. ∎

Remark 2.

In Lemma 3.2, Theorem A and the corollary, if we replace “every non-central element x of G” by “every non-central element x of order divisible by at most two distinct primes”, then, by similar arguments, we can prove that the conclusions of Lemma 3.2, Theorem A and the corollary still hold.

Question.

It is an interesting topic to consider the centralizers of prime power order elements, namely, for every non-central prime power order element x of a finite group G, if CG(x)G, what is the structure of G?


Communicated by Evgenii I. Khukhro


Award Identifier / Grant number: 11771271

Award Identifier / Grant number: 11901169

Funding source: Henan Normal University

Award Identifier / Grant number: YJS2019JG06

Funding statement: The research of the work was supported by the National Natural Science Foundation of China (11771271, 11901169), the project for high quality courses of postgraduate education in Henan Province, research and practice project of higher education reform in Henan Normal University (post-graduate education, No. YJS2019JG06).

Acknowledgements

The authors would like to thank the referee for their valuable suggestions and useful comments that contributed to the final version of this paper.

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Received: 2019-10-15
Revised: 2020-02-07
Published Online: 2020-03-10
Published in Print: 2020-09-01

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