The Normalizer Property for Integral Group Rings of Finite Solvable T-Groups

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J. Group Theory, Ahead of Print DOI 10.1515/JGT.2011.105 © de Gruyter 2011

The normalizer property for integral group rings of ﬁnite solvable T-groups

Zhengxing Li and Jinke Hai Communicated by John S. Wilson

Abstract. A group is said to be a T-group if all its subnormal subgroups are normal. Let G be a ﬁnite solvable T-group. It is shown that the normalizer property holds for G. As a direct consequence of our result, we obtain that the normalizer property holds for ﬁnite groups all of whose Sylow subgroups are cyclic.

1 Introduction

Let G be a finite group and let ZG be its integral group ring over Z. Denote by U.ZG/ the group of units of ZG. A longstanding problem (see [21, Prob- lem 43]) asks whether N .G/ G Z.U.ZG// for any finite group G, where U.ZG/ D Z.U.ZG// is the center of U.ZG/ and NU.ZG/.G/ is the normalizer of G in U.ZG/. If this equality holds for G, then we will say that G has the normalizer property. This equality was first confirmed for finite nilpotent groups by Cole- man [1], and later this result was extended to any finite group having a normal Sylow 2-subgroup by Jackowscki and Marciniak [7]. It was Mazur [14–16] who first noticed that there are close connections between the normalizer problem and the isomorphism problem. Based on Mazur’s observations, among other things, Hertweck [2] constructed counter-examples for both the normalizer problem and the isomorphism problem. Nevertheless, it is still of interest to determine for which groups the normalizer property holds. Recently, many positive results on the normalizer problem have been obtained by many authors. For instance, Li, Sehgal and Parmenter [12] proved that the normalizer property holds for finite Blackburn groups. Petit Lobão and Polcino Mi- lies [17] confirmed that the normalizer property holds for finite Frobenius groups. Petit Lobão and Sehgal [18] showed that the normalizer property holds for complete monomial groups. In addition, other positive results on this problem appeared in [3–5, 8, 10–12].

Supported by the National Natural Science Foundation of China (Grant No. 11071155 and Grant No. 11171169) and the Natural Science Foundation of Shandong Province (Grant No. Y2008A03).

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The normalizer problem may be restated in an equivalent form in terms of a class of special automorphisms of ﬁnite groups. For any u NU.ZG/.G/, write 2 1 u for the automorphism of G induced by u via conjugation, i.e. u.g/ ugu D for all g G. All such automorphisms of G form a subgroup of Aut.G/, denoted 2 by AutZ.G/. Obviously Inn.G/ 6 AutZ.G/:

A question (see [7, Question 3.7]) asks whether AutZ.G/ Inn.G/ for any finite D group G. This question has a negative answer in general. Hertweck [2] established the existence of a metabelian group G of order 225 972 which has a non-inner automorphism induced by a unit in NU.ZG/.G/ of argument 1 (for details, see [2, Theorem A] and its proof). It is easy to see that this question is equivalent to the normalizer problem. The aim of this article is to investigate the normalizer problem for integral group rings of finite solvable T-groups by using the equivalent form mentioned above. Recall that a group G is said to be a T-group if every subnormal subgroup of G is normal. It is known that finite solvable T-groups are metabelian. For general information on the structure of finite solvable T-groups, we refer the reader to [20]. As mentioned above, Hertweck’s result ([2, Theorem A]) tells us that the normalizer problem fails to hold for all metabelian groups. So it is of interest to know for which class of metabelian groups the normalizer property holds. In this direction, some positive results can be found in [9] and [13]. In this paper, we confirm that finite solvable T-groups, as a class of metabelian groups, also have the normalizer property. Our main result is as follows.

Theorem A. Suppose that G is a finite solvable T-group. Then the normalizer property holds for G. All groups considered are finite. Let N be a subgroup of G and let Aut.G/. 2 We write N for the restriction of to N . Suppose additionally that N E G and j fixes N . By abuse of notation, we write for the automorphism of G=N jG=N induced by . Let g be a fixed element in G. We write conj.g/ for the inner automorphism of G induced by g via conjugation. Denote by .G/ the set of all primes dividing G . Other notation will be mostly standard (as in [20]). j j 2 Preliminaries

In this section, we collect some preliminary results which will be used in the proof of Theorem A.

Lemma 2.1. Let G be a ﬁnite group such that all central units of ZG are trivial. Then NU.ZG/.G/ consists of trivial units.

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The normalizer property for integral group rings 3

Proof. Let u be a unit of ZG which normalizes G. Then gu2 is central for some g G. Since central units are trivial, u2 G. Thus u is of ﬁnite order. A theorem 2 2 of Passman implies that u is a trivial unit (see also [16, Lemma 8]).

Note that a characterization of ﬁnite groups whose integral group rings have only trivial central units had been obtained by Ritter and Sehgal (see [19, Theo- rem]).

Lemma 2.2. Let G be a ﬁnite group and let u ZG be an element with augmen- 2 tation 1. Let be an automorphism of G such that u .g/ug 1 for all g G. D 2 Then for any Sylow subgroup P of G there is an element g in the support of u such that coincides with conjugation by g on P .

Deﬁnition 2.3 ([6]). Let G be a ﬁnite group and Aut.G/. Then is called a 2 Coleman automorphism of G if the restriction of to any Sylow subgroup P of G equals the restriction of some inner automorphism of G. All such automorphisms of G form a subgroup of Aut.G/, denoted by AutCol.G/.

Lemma 2.4 ([6, Proposition 1]). The prime divisors of AutCol.G/ lie in .G/, j j the set of prime divisors of G . j j

Lemma 2.5 (Krempa, [20, Proposition 9.5]). Let u N .G/ and let u be 2 U.ZG/ the automorphism of G induced by u via conjugation. Then 2 Inn.G/. u 2 Lemma 2.6 (Gaschütz, [20, (13.4.4) ]). Let G be a ﬁnite solvable T -group and let L ŒG ; G. Then L is the smallest term of the lower central series, L is an abel- WD 0 ian group of odd order and G=L is a Dedekind group.

Lemma 2.7 (Dedekind, Baer, [20, (5.3.7)]). Let G be a ﬁnite group. Then G is a Dedekind group if and only if G is abelian or the direct product of a quaternion group of order 8, an elementary abelian 2-group and an abelian group of odd order.

Lemma 2.8 ([13, Theorem 12.3]). Let G be a ﬁnite metabelian group with an abelian Sylow 2-subgroup. Then the normalizer property holds for G.

Lemma 2.9 (Higman, [21, Theorem 2.7]). Let G be a ﬁnite group. Then ZG has only trivial units if and only if G is an abelian group of exponent 2; 3; 4; 6 or G Q8 E, where Q8 is a quaternion group of order 8 and E is an elementary Š abelian 2-group.

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3 Proof of Theorem A

In this section, we shall give a proof for Theorem A. We begin with the following result, which is due to the referee, as is its proof. It is appropriate here to thank the referee for this, and also for providing Theorem 3.2 and its proof so that our main result (Theorem A) goes through immediately as a consequence of Theorem 3.2, among other lemmas.

Proposition 3.1. Let G be a ﬁnite group with a normal subgroup A such that central units of Z.G=A/ are trivial. Let AutZ.G/. Then there is an element 2 b G such that for any Sylow subgroup P of G there exists an a A such that 2 2 the restriction of to P coincides with conjugation by ba.

Proof. Let H G=A and let s H G be a section such that s.eH / eG, D W ! D where eM is the identity element of a group M . Let u be a unit of ZG inducing via conjugation. We may assume that the augmentation .u/ 1. Then, by Lem- D 1 ma 2.1, the image u of u in ZH is an element t of H. Let b s.t/, w b u and 1 N 1 D D let conj.b /. Then w 1 and w .g/wg for any g G. We may D P N D D P2 write w h H s.h/wh for some wh ZA. We have 1 w h H .wh/h. D 2 2 DN D 2 It follows that .ue/ 1 and .u / 0 for all h e, where e eH . D h D ¤ D Let g G and let g be its image in H . Then for any h H there exists ch A 2 N1 1 2 1 2 such that .g/s.h/g s.ghg /c . In particular, ce .g/g . We have D N N h D X X s.h/w w .g/wg 1 s.ghg 1/c gw g 1: h D D D N N h h h H h H 2 2 1 1 Note that chgwhg ZA for all h H. It follows that wghg 1 chgwhg 2 2 1 N N D 1 for all h H. Taking h e, we get we cegweg .g/weg . Let P 2 D D D be a Sylow subgroup of G. Then, by Lemma 2.2, there exists a A such that 2 coincides with conjugation by a on P . This means that acts on P as conjugation by ba. This proves Proposition 3.1.

Theorem 3.2. Let G be a ﬁnite group with a normal subgroup A of odd order such that central units of Z.G=A/ are trivial. Then AutZ.G/ Inn.G/. D Proof. Let AutZ.G/. Then, by Proposition 3.1, there exists b G such that 2 2 the automorphism conj.b 1/ acts as conjugation by some element in A on D every Sylow subgroup of G. It follows that , when restricted to A, is a Coleman automorphism of A and induces the identity on G=A. Since, by Lemma 2.5, 2 is inner, we may assume that the order of is a power of 2. Then, by Lemma 2.4, is trivial on A. Now, if g G, then .g/ ag for some a A. Then we get 2 D 2 n.g/ ang for all positive integers n. Since the order of a is odd and the order D

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The normalizer property for integral group rings 5 of is a power of 2, we have a 1. This proves that is the identity on G. D It follows that Inn.G/. As is arbitrary, we obtain that AutZ.G/ Inn.G/. 2 Â On the other hand, we have Inn.G/ AutZ.G/. Hence, AutZ.G/ Inn.G/, as Â D desired.

Proof of Theorem A. Let G be a ﬁnite solvable T-group and write L ŒG ; G. WD 0 Then, by Lemma 2.6, L is an abelian group of odd order and G=L is a Dedekind group. According to Lemma 2.7, we divide the proof into two cases. Case 1. G=L is abelian. Note that G is a metabelian group in this case. In addition, every Sylow 2-subgroup of G is abelian since L is of odd order and G=L is abelian. Hence, by Lem- ma 2.8, the normalizer property holds for G. Case 2. G=L is nonabelian. In this case, by Lemma 2.7, we may assume that G=L Q8 A E, where D Q8 is a quaternion group of order 8, A is an abelian group of odd order and E is an elementary abelian 2-group. Since A is normal in G=L, it follows that there exists a normal subgroup M of G with L 6 M such that A M=L. Since A and L are D of odd order and M A L , it follows that M is also of odd order. Note that j j D j jj j G=M .G=L/=.M=L/ Q8 E, i.e. G is an extension of a group of odd order Š Š by a Hamilton 2-group. Since, by Lemma 2.9, the integral group ring Z.G=M / has only trivial units, it follows from Theorem 3.2 that AutZ.G/ Inn.G/, i.e. the D normalizer property holds for G. This completes the proof of Theorem A.

Finally, we record two simple corollaries of Theorem A as follows.

Corollary 3.3. Let G be a ﬁnite solvable T-group. Then the normalizer property holds for any subgroup of G.

Proof. It is known that any subgroup of a ﬁnite solvable T-group is still a ﬁnite solvable T-group. So the assertion follows directly from Theorem A.

Corollary 3.4. Let G be a ﬁnite group all of whose Sylow subgroups are cyclic. Then the normalizer property holds for G.

Proof. Since, by assumption, the Sylow subgroups of G are cyclic, it follows that the given G is a solvable T-group and thus the assertion follows immediately from Theorem A.

Acknowledgments. The authors thank the referee for his/her valuable advice and suggestions, and also for providing Proposition 3.1 and Theorem 3.2 and their proofs so that our main result follows easily as a consequence of Theorem 3.2.

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Received May 14, 2011; revised July 21, 2011.

Author information Zhengxing Li, College of Mathematics, Qingdao University, Qingdao 266071, P.R. China. E-mail: [email protected] Jinke Hai, College of Mathematics, Qingdao University, Qingdao 266071, P.R. China. E-mail: [email protected]

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