# Class-preserving Coleman automorphisms of some classes of finite groups

• Jingjing Hai , Zhengxing Li and Xian Ling
From the journal Open Mathematics

## Abstract

The normalizer problem of integral group rings has been studied extensively in recent years due to its connection with the longstanding isomorphism problem of integral group rings. Class-preserving Coleman automorphisms of finite groups occur naturally in the study of the normalizer problem. Let G be a finite group with a nilpotent subgroup N . Suppose that G / N acts faithfully on the center of each Sylow subgroup of N . Then it is proved that every class-preserving Coleman automorphism of G is an inner automorphism. In addition, if G is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of G is an inner automorphism. Other similar results are also obtained in this article. As direct consequence, the normalizer problem has a positive answer for such groups.

MSC 2010: 20C05; 20E36; 16S34

## 1 Introduction

All groups considered in this article are finite. Let M be a subgroup of G and let σ Aut ( G ) . We write σ M for the restriction of σ to M . Furthermore, suppose that M G and σ fixes M . Then, by abuse of notation, we write σ G / M for the automorphism of G / M induced by σ . 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 π ( G ) the set of all primes dividing G . Other notations will be mostly standard, refer to [1,2].

Let G be a finite group and Z G be its integral group ring over Z . Denote by U ( Z G ) the group of units of Z G . The normalizer problem (see problem 43 in [2]) of integral group rings asks whether N U ( Z G ) ( G ) = G Z ( U ( Z G ) ) for any finite group G , where N U ( Z G ) ( G ) and Z ( U ( Z G ) ) denote the normalizer of G in U ( Z G ) and the center of U ( Z G ) , respectively. If the equality is valid for G , then we say that the normalizer property holds for G .

This equality was first shown to be true for finite nilpotent groups by Coleman in [3], and later this result was extended to any finite group having a normal Sylow 2-subgroup by Jackowski and Marciniak in [4]. It was Mazur who first noted that there are close connections between the normalizer problem and the isomorphism problem (see [5,6,7]). Based on Mazur’s observations, among other things, Hertweck in [8] constructed the first counterexample to the normalizer problem and then the first counterexample to the isomorphism problem. Nevertheless, it is still of interest to determine for which groups the normalizer property holds. Recently, lots of positive results on the normalizer problem can be found in [9,10,11, 12,13].

For any u N U ( Z G ) ( G ) , we write φ u to denote the automorphism of G induced by u via conjugation, i.e., g φ u = u 1 g u for all g G . All such automorphisms of G form a subgroup of Aut ( G ) , denoted by Aut Z ( G ) . It is not hard to see that Inn ( G ) Aut Z ( G ) . Let Out Z ( G ) Aut Z ( G ) / Inn ( G ) . A question (see Question 3.7 in [4]) asks whether Out Z ( G ) = 1 for any finite group G .

It turns out that the aforementioned question is equivalent to the normalizer problem. Due to this, it is not a surprise that some classes of special automorphisms occur naturally in the study of the normalizer problem. Aut c ( G ) denotes the class-preserving automorphism group of G , in which every automorphism sends g G to some conjugate of g . Aut Col ( G ) denotes the Coleman automorphism group of G , in which the restriction of every automorphism to each Sylow subgroup of G equals the restriction of some inner automorphism of G . Set Out c ( G ) = Aut c ( G ) / Inn ( G ) and Out Col ( G ) = Aut Col ( G ) / Inn ( G ) . It is known by Coleman’s lemma (see in [3]) that Out Z ( G ) Out c ( G ) Out Col ( G ) . In addition, Krempa showed that Out Z ( G ) is an elementary abelian 2-group (proof can be found in [4]). Thus, if one can show that Out c ( G ) Out Col ( G ) is of odd order, then Out Z ( G ) = 1 , namely, the normalizer property holds for such group G .

In this direction, Hertweck (see [1,12]) proved that if Sylow 2-subgroups of a finite group G are cyclic, dihedral, or generalized quaternion, then Out c ( G ) Out Col ( G ) is of odd order. Marciniak and Roggenkamp [9] proved that the normalizer property holds for metabelian groups with abelian Sylow 2-subgroups. For other related results, see [14,15,16,17, 18,19].

The aim of this article is to investigate class-preserving Coleman automorphisms of some classes of finite groups without any restrictions on the structure of Sylow 2-subgroups. In Section 2, we present some lemmas which will be used in the sequel. In Section 3, we give some results on class-preserving Coleman automorphisms of some groups. Particularly, we prove that if G is a finite group with a nilpotent normal subgroup N and G / N acts faithfully on the center of each Sylow subgroup of N , then Out c ( G ) Out Col ( G ) = 1 . The counterexample to the normalizer problem constructed by Hertweck (see[8]) is a metabelian group. However, we can show that if G is the product of a cyclic normal subgroup and an abelian subgroup, then Out c ( G ) Out Col ( G ) = 1 ; in particular, the normalizer property holds for G . Some other related results are also obtained in Section 3.

## 2 Preliminaries

In this section, some lemmas needed in the sequel are presented.

## Lemma 2.1

[17] Let G be a finite group with a nilpotent normal subgroup N. Assume that P is an arbitrary Sylow subgroup of N and G / N acts faithfully on Z ( P ) . Then C G ( P ) N . In particular, C G ( N ) N .

## Lemma 2.2

[19] Let P be a normal p-subgroup of a finite group G . If C G ( P ) P , then G has no noninner p-central automorphisms. In particular, Out Col ( G ) = 1 .

## Lemma 2.3

Let G be a finite group, H be a subgroup of G, and let σ be an automorphism of G of p -power order, where p is a prime. If there is x G such that σ H = conj ( x ) H , then there exists some γ Inn ( G ) such that σ γ H = i d H and σ γ is still of p-power order.

## Proof

Set o ( σ ) = p i , where i N . Write β conj ( x ) . Then σ H = β H , i.e., σ β 1 H = i d H . Let n N such that ( σ β 1 ) n be the p -part of σ β 1 with ( n , p ) = 1 . Then there exists s , t Z such that s n + t p i = 1 . Obviously, ( σ β 1 ) s n is of p -power order and ( σ β 1 ) s n H = i d H . Note that Inn ( G ) Aut ( G ) , so there exists some γ Inn ( G ) such that ( σ β 1 ) s n = σ s n γ = σ 1 t p i γ = σ γ . Hence, γ is the desired inner automorphism.□

## Lemma 2.4

[1] Let p be a prime, and σ an automorphism of G of p-power order. Assume further that there is N G such that σ fixes all elements of N, and that σ induces the identity on G / N . Then σ induces the identity on G / O p ( Z ( N ) ) . If σ fixes in addition a Sylow p-subgroup of G element-wise, then σ is an inner automorphism.

## Lemma 2.5

[1] Let N G and let p be a prime which does not divide the order of G / N . Then the following hold.

1. If σ Aut ( G ) is a class-preserving or Coleman automorphism of G of p -power order, then σ induces a class-preserving or a Coleman automorphism of N , respectively;

2. If Out c ( N ) or Out Col ( N ) is a p -group, then so is Out c ( G ) or Out Col ( G ) . If Out c ( N ) Out Col ( N ) is a p -group, then so is Out c ( G ) Out Col ( G ) .

## Lemma 2.6

Let G be a finite group and let N be a subgroup of G . Let σ be an automorphism of G of p -power order with p a prime. Suppose that σ fixes N and σ N = conj ( x ) N for some x G . Then there exists a p -element y G such that σ N = conj ( y ) N .

## Proof

Let o ( σ ) = p i , o ( x ) = p j t , where i , j , t N and ( p , t ) = 1 . Set k = max { i , j } . Since ( p k , t ) = 1 , it follows that there exists u , v Z such that u p k + v t = 1 . Write y = x v t . Then it is obvious that y is a p -element. For any z N , since z = z σ u p k = z x u p k , it follows that z σ = z x = z x u p k + v t = ( z x u p k ) x v t = z x v t = z y , namely, σ N = conj ( y ) N .□

## Lemma 2.7

[20] Let A be an abelian p -group and let N be a p-group on which A acts, where p is a prime. Then C A ( N ) = C A ( x ) = C A ( N ¯ ) = C A ( x ¯ ) for some x N , where N ¯ = N / Φ ( N ) with Φ ( N ) being the Frattini subgroup of N .

## Theorem 3.1

Let G be a finite group with a nontrivial nilpotent normal subgroup N. Assume that G / N acts faithfully on the center of each Sylow subgroup of N. Then Out c ( G ) Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

## Proof

Let q π ( G ) and let σ Aut c ( G ) Aut Col ( G ) be of q -power order. We have to show that σ Inn ( G ) . If π ( N ) = 1 , then by Lemma 2.1, C G ( N ) N . Furthermore, by Lemma 2.2, this implies that Out Col ( G ) = 1 . In particular, Out c ( G ) Out Col ( G ) = 1 . Hereafter, we assume that π ( N ) = r > 1 .

Claim 1. σ N = i d N .

Let π ( N ) = { p 1 , p 2 , , p r } and let P i Syl p i ( N ) , where i = 1 , 2 , , r . Then N = P 1 × P 2 × × P r . Since σ Aut Col ( G ) , there exists some h i G such that

(1) σ P i = conj ( h i ) P i .

For any z i Z ( P i ) , by equation (1), we have

(2) z i σ = h i 1 z i h i .

For any z i Z ( P i ) and z j Z ( P j ) with i j , by equation (2),

(3) ( z i z j ) σ = h i 1 z i h i h j 1 z j h j .

On the other hand, since σ Aut c ( G ) , there exists h G such that

(4) ( z i z j ) σ = h 1 z i h h 1 z j h .

Combining equation (3) with (4), we obtain

(5) ( h i 1 z i h i ) ( h j 1 z j h j ) = ( h 1 z i h ) ( h 1 z j h ) .

Since N is nilpotent, by equation (5), we have h i 1 z i h i = h 1 z i h and h j 1 z j h j = h 1 z j h . That is,

(6) ( h i h 1 ) 1 z i ( h i h 1 ) = z i ,

(7) ( h j h 1 ) 1 z j ( h j h 1 ) = z j .

Since h ¯ i h ¯ i 1 is cyclic, there exists some element in Z ( P i ) , say z i , such that

C h ¯ i h ¯ i 1 ( z i ) = z Z ( P i ) C h ¯ i h ¯ i 1 ( z ) = C h ¯ i h ¯ i 1 ( Z ( P i ) ) C G / N ( Z ( P i ) ) = 1 .

So by equations (6) and (7), h N = h i N = h j N . As i , j are arbitrary, we have h N = h 1 N = h 2 N = = h r N . Set h i = h n i with n i N , i = 1 , 2 , , r . For any x i P i , by equation (1),

(8) x i σ = n i 1 h 1 x i h n i .

As N is nilpotent, we may assume n i P i in equation (8). Write n n 1 n 2 n r . Then, by equation (8), for any x = x 1 x 2 x r N with x i P i ,

(9) x σ = n 1 h 1 x h n .

This shows that σ N = conj ( h n ) N . By Lemma 2.3, we may assume that σ N = i d N , as claimed.

Claim 2. σ G / N = i d G / N .

For any g G and n N , by Claim 1, n g = ( n g ) σ = n g σ , implying g σ g 1 C G ( N ) . Recall that C G ( N ) N . So the preceding equality implies that σ G / N = i d G / N , as claimed.

Claim 3. σ Inn ( G ) .

By Lemma 2.4, Claims 1 and 2 yield that σ G / O q ( Z ( N ) ) = i d G / O q ( Z ( N ) ) . If q π ( N ) , then the preceding equation implies that σ = i d . It remains to consider the case q π ( N ) . Let Q be a Sylow q -subgroup of G fixed by σ . Then Q 1 Q N is the Sylow q -subgroup of N . Since σ Aut Col ( G ) , there exists some q -element g G such that σ Q = conj ( g ) Q . In particular, σ Z ( Q 1 ) = conj ( g ) Z ( Q 1 ) . On the other hand, By Claim 1, σ Z ( Q 1 ) = i d Z ( Q 1 ) . Consequently, g C G ( Z ( Q 1 ) ) . It follows that g N C G / N ( Z ( Q 1 ) ) . From this we deduce that g N = N since the action of G / N on Z ( Q 1 ) is faithful. So g N and hence g Q 1 . Note that σ Q 1 = conj ( g ) Q 1 = i d Q 1 . So g Z ( Q 1 ) Z ( N ) . It follows that σ conj ( g 1 ) N = i d N , σ conj ( g 1 ) G / N = i d G / N , and σ conj ( g 1 ) Q = i d Q . So by Lemma 2.4 σ Inn ( G ) . We are done.□

As immediate consequences of Theorem 3.1, we have the following results.

## Corollary 3.2

Let G = N w r K be the standard wreath product of N by K, where N is a nontrivial nilpotent group and K is an arbitrary group. Then Out c ( G ) Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

## Proof

Let K = r . Then G = N w r K = N r K , where N r is the direct product of r copies of N . Let p π ( N ) and let P Syl p ( N r ) . We will show that K acts faithfully on Z ( P ) . Since N is a nilpotent group, thus N r is also a nilpotent group. Obviously, K acts on Z ( P ) . For any y Z ( P ) , if y h = y , where h K . Since the intersection of Z ( P ) with each component of N r is nontrivial, i.e., Z ( P ) is extensive in N r , we deduce that h = 1 , this shows that K acts faithfully on Z ( P ) . Thus, the assertion follows from Theorem 3.1.□

As a direct consequence of Corollary 3.2, we have the following result, which generalizes a well-known result due to Petit Lobão and Sehgal ([11], Theorem 1).

## Corollary 3.3

Let G = N w r S m , where N is a finite nilpotent group and S m is the group of all permutations on m letters. Then the normalizer property holds for G.

## Corollary 3.4

Let G be the holomorph of an arbitrary nilpotent group N , i.e., G = N Aut ( N ) . Then Out c ( G ) Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

## Theorem 3.5

Let G = N A , where N is a nilpotent normal subgroup and A is an abelian subgroup. Then Out c ( G ) Out Col ( G ) is a p -group for each p π ( G ) π ( N ) π ( A ) .

## Proof

For any p π ( G ) π ( N ) π ( A ) , let ρ Aut c ( G ) Aut Col ( G ) be of p -power order. We will show that ρ is an inner automorphism.

Case 1. p π ( N ) .

Let N p be the Sylow p -subgroup of N . Then N p char N G . It follows that N p is a normal Sylow p -group of G . By Lemma 2.5(2) (replacing N therein with N p ), Out Col ( G ) is a p -group. In particular, Out c ( G ) Out Col ( G ) is a p -group.

Case 2. p π ( A ) .

Let A p be the Sylow p -subgroup of A . Then N A p is normal in G since G / N is abelian. Note that G / N A p is a p -group. By Lemma 2.5, we may assume that G = N A p .

Claim 1. ρ conj ( g 1 ) N Aut Col ( N ) for some g G .

Let π ( N ) = { p 1 , p 2 , , p r } and let P i Syl p i ( N ) , where i = 1 , 2 , , r . Then N = P 1 × P 2 × × P r . Since ρ is a Coleman automorphism, by Lemma 2.6, for each P i , there exists a p -element h i G such that

(10) ρ P i = conj ( h i ) P i .

By Lemma 2.7, C A p ( P i ) = C A p ( x i ) for some x i P i , where i = 1 , 2 , , r . Write x = x 1 x 2 x r . Then there exists g G such that x ρ = x g , i.e., ( g 1 x 1 g ) ( g 1 x r g ) = ( h 1 1 x 1 h 1 ) ( h r 1 x r h r ) . From this we obtain g 1 x i g = h i 1 x i h i . It follows that

(11) [ h i g 1 , x i ] = 1 .

Since G = N A p , we may set h i g 1 = n a , where n N and a A p . We will show a C A p ( P i ) . Let P i ¯ P i / Φ ( P i ) , where Φ ( P i ) is the Frattini subgroup of P i . Then G acts on P i ¯ . By equation (11), we have

(12) [ h i g 1 , x ¯ i ] = 1 .

On the other hand,

(13) [ h i g 1 , x ¯ i ] = [ n a , x ¯ i ] = [ a , x ¯ i ] .

So equations (12) and (13) imply that [ a , x ¯ i ] = 1 . Again by Lemma 2.7, a C A p ( x ¯ i ) = C A p ( P i ) . This, together with equation (10), implies that ρ conj ( g 1 ) P i = conj ( n ) P i . This shows that ρ conj ( g 1 ) N Aut Col ( N ) , as claimed.

Claim 2. ρ Inn ( G ) .

Since N is nilpotent, it follows that ρ conj ( g 1 ) N = conj ( n ) N for some n N . That is, ρ N = conj ( n g ) N . With this in hand, by Lemma 2.3, we may assume that

(14) ρ N = i d N .

Note that G / N is abelian. So we have

(15) ρ G / N = i d G / N .

By Lemma 2.4, equations (14) and (15) yield that ρ G / O p ( Z ( N ) ) = i d G / O p ( Z ( N ) ) . Note that N is a p -group. So the preceding equality is precisely ρ = i d . We are done.□

## Corollary 3.6

Let G be an extension of a p-group by an abelian group. Then Out c ( G ) Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

## Corollary 3.7

Let G be an extension of a nilpotent group of odd order by an abelian group. Then Out c ( G ) Out Col ( G ) is of odd order. In particular, the normalizer property holds for G.

Marciniak and Roggenkamp (see [9]) constructed a finite metabelian group G = ( C 2 4 × C 3 ) C 2 3 for which Out c ( G ) Out Col ( G ) is of even order. It is clear that the group G is the semidirect product of a cyclic group of 3 by a nonabelian 2-group. This shows that if G is the product of a cyclic normal subgroup and a nilpotent subgroup, then it is not necessary that Out c ( G ) Out Col ( G ) is trivial. However, we can prove the following result.

## Theorem 3.8

Let G = C A , where C is a cyclic normal subgroup and A is an abelian subgroup. Then Out c ( G ) Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

## Proof

Let p π ( G ) and ρ Aut c ( G ) Aut Col ( G ) be of p -power order. We will show that ρ is inner. If either p π ( C ) π ( A ) or p π ( A ) π ( C ) , then by Theorem 3.5 ρ Inn ( G ) . If p π ( C ) π ( A ) , then by Lemma 2.5(2) we may assume that A itself is a p -subgroup. Since C is cyclic and ρ Aut c ( G ) , it follows that there exists some g G such that ρ C = conj ( g ) C . Without loss of generality, we may set ρ C = i d C . Let P C be the Sylow p -subgroup of C . Then P = P C A is a Sylow p -subgroup of G . Without loss of generality, we may assume that ρ fixes P and ρ P = conj ( x ) P for some p -element x . Set x = a b with a P C and b A . Note that i d P C = ρ P C = conj ( a b ) P C = conj ( b ) P C . This yields that b C A ( P C ) . Since A is abelian, it follows that b Z ( P ) and thus ρ P = conj ( x ) P = conj ( a b ) P = conj ( a ) P . From this we deduce that ρ = conj ( a ) . We are done.□

## Acknowledgements

The author wishes to thank the referees for their useful comments. The work was partially supported by the National Natural Science Foundation of China (Grant No. 11871292).

1. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

2. Conflict of interest: The authors state no conflict of interest.

## References

[1] M. Hertweck, Class-preserving Coleman automorphisms of finite groups, Monatsh. Math. 144 (2002), no. 1, 1–7, https://doi.org/10.1007/s006050200029. Search in Google Scholar

[2] S. K. Sehgal, Units in Integral Group Rings, Longman Scientific and Technical Press, New York, 1993. Search in Google Scholar

[3] D. B. Coleman, On the modular group ring of a p-group, Proc. Amer. Math. Soc. 5 (1964), no. 4, 511–514, https://doi.org/10.2307/2034735. Search in Google Scholar

[4] S. Jackowski and Z. S. Marciniak, Group automorphisms inducing the identity map on cohomology, J. Pure Appl. Algebra 44 (1987), no. 3, 241–250, https://doi.org/10.1016/0022-4049(87)90028-4. Search in Google Scholar

[5] M. Mazur, The normalizer of a group in the unit group of its group ring, J. Algebra 212 (1999), no. 1, 175–189, https://doi.org/10.1006/jabr.1998.7629. Search in Google Scholar

[6] M. Mazur, Automorphisms of finite groups, Comm. Algebra 22 (1994), no. 15, 6259–6271, https://doi.org/10.1080/00927879408825187. Search in Google Scholar

[7] M. Mazur, On the isomorphism problem for infinite group rings, Expo. Math. 13 (1995), no. 13, 433–445, https://doi.org/0723-0869/95/050433-445. Search in Google Scholar

[8] M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math. (2) 154 (2001), no. 1, 115–138, https://doi.org/10.2307/3062112. Search in Google Scholar

[9] Z. S. Marciniak and K. W. Roggenkamp, The normalizer of a finite group in its integral group ring and Čech cohomology, in: Algebra-Representation Theory, NATOASI Series II, Vol. 28, Kluwer Academic, Dordrecht, 2001, pp. 159–188. 10.1007/978-94-010-0814-3_8Search in Google Scholar

[10] Y. Li, The normalizer property of a metabelian group in its integral group ring, J. Algebra 256 (2002), no. 2, 343–351, https://doi.org/10.1016/S0021-8693(02)00102-3. Search in Google Scholar

[11] T. Petit Lobão and S. K. Sehgal, The normalizer property for integral group rings of complete monomial groups, Comm. Algebra 31 (2003), no. 6, 2971–2983, https://doi.org/10.1081/AGB-120021903. Search in Google Scholar

[12] M. Hertweck, Local analysis of the normalizer problem, J. Pure Appl. Algebra 163 (2001), no. 3, 259–276, https://doi.org/10.1016/S0022-4049(00)00167-5. Search in Google Scholar

[13] Z. Li, Y. Li, and H. Gao, The influence of maximal quotient groups on the normalizer conjecture of integral group rings, J. Group Theory 19 (2016), no. 6, 983–992, https://doi.org/10.1515/jgth-2016-0013. Search in Google Scholar

[14] S. O. Juriaans, J. M. Miranda, and J. R. Robério, Automorphisms of finite groups, Comm. Algebra 32 (2004), no. 5, 1705–1714, https://doi.org/10.1081/AGB-120029897. Search in Google Scholar

[15] M. Hertweck and W. Kimmerle, Coleman automorphisms of finite groups, Math. Z. 242 (2002), no. 3, 203–215, https://doi.org/10.1007/s002090100318. Search in Google Scholar

[16] M. Hertweck, Class-preserving automorphisms of finite groups, J. Algebra 241 (2001), no. 1, 1–26, https://doi.org/10.1006/jabr.2001.8760. Search in Google Scholar

[17] J. Hai, Coleman automorphisms of finite groups with a self-centralizing normal subgroup, Czech Math. J. 70 (2020), no. 4, 1197–1204, https://doi.org/10.21136/CMJ.2020.0423-19. Search in Google Scholar

[18] Z. Li, Coleman automorphisms of permutational wreath products II, Comm. Algebra 46 (2018), no. 10, 4473–4479, https://doi.org/10.1080/00927872.2018.1448836. Search in Google Scholar

[19] A. Van Antwerpen, Coleman automorphisms of finite groups and their minimal normal subgroups, J. Pure Appl. Algebra 222 (2018), no. 11, 3379–3394, https://doi.org/10.1016/j.jpaa.2017.12.013. Search in Google Scholar

[20] M. Hertweck and E. Jespers, Class-preserving automorphisms and the normalizer property for Blackburn groups, J. Group Theory 12 (2009), no. 1, 157–169, https://doi.org/10.1515/JGT.2008.068. Search in Google Scholar