Coleman Automorphisms of Some Metabelian Groups

Abstract

In this study, the approach is through the projective limits, and we obtain some characterizations of the Coleman automorphism groups of a split metabelian group and a generalized dicyclic group. An application of our results is a generalization of the result of Z. Li and Y. Li.

1. Introduction

Throughout this study, all groups considered are finite and denotes the automorphism group of a finite group , and we use to denote the inner automorphism on . Then, we have the following two subgroups of .

consists of those , such that for any prime and any Sylow -subgroup , there exists a , such that .

, where is the integral group ring, and is the normalizer of in the unit group of . We set , .

Recall that Coleman automorphisms come up in the study of the normalizer problem for integral group rings of finite groups, i.e., does hold, where is the center of . It is easy to see that is equivalent to . Coleman’s lemma [1] implies that . In [2], Krempa proved that is a 2-group. Thus, if we show that is a group of odd order or under some conditions, then , that is, the normalizer problem holds for . Related results on this subject can be found in [3–9].

The purpose of this study is to determine the structure of the Coleman automorphism groups of some metabelian groups. Li and Li [10] obtained a characterization of of a generalized dihedral group . In this study, the Coleman automorphism group of a split metabelian group is characterized by using the projection limit property; this result is an extension and alternative proof of Theorem 1.2 in [10]. We also give the structure of of a generalized dicyclic group . Our notation is mostly standard, refer to [2, 11, 12].

2. Preliminaries

Definition 1. Let be a finite group, and let . Put , and assume that is the natural homomorphisms, where is a natural number. Then, the projective limit of the quotients with respect to the natural maps is the subgroupof the direct product of the quotients ([13], Page 115, Definitions).

Lemma 1 (See [14]). Let be a finite group, and let . Assume that(1)(2)For any prime , there exists a , such that . Then, .

Lemma 2 (See [13]). Let be a finite solvable group. Then,where denotes the largest normal -subgroup of .

Lemma 3. Let be a finite solvable group. Put , where . Then,(1)The natural maps induce maps , where is the center of and is the center of , respectively.(2)

Proof. (1)Set It easily is verified that is a surjective homomorphism(2)By Lemma 1, we obtain . For any , then Definition 1 implies that . SetThen, are the isomorphisms. Assume thatIt is clear that are the surjective homomorphisms. First, we show thatIn fact,On the other hand,LetIt easily is verified that is an isomorphism.
Next, we shall show thatSince , so it remains to show thatBy Definition 1 and Lemma 1, . Thus, for any with , , writeIt follows thatthat is, . Moreover, it easily is verified that . By Lemma 2, we obtain , i.e., .
Conversely, suppose that , then by Lemma 2, . Write . Consequently,For any with , we have ; it follows that , that is, , which implies that . Hence,

3. Proof of the Theorems

Theorem 1. Let be a split metabelian group, where for every , , and denotes the exponent of . Then, if , then , and if , then .

Proof. (1)If , then is a -group and (2)If , let , where are distinct odd primes and  Case 1:  Since , it follows that , Write , then . Thus, . By Lemma 3, Case 2:  Since , . It follows that , Write . Thus, , which implies that . By Lemma 3, Case 3: Let , where . Since , then is of order . Thus,Write , we have ,which implies that is of order , . By Lemma 3,Let be an abelian group and be an involution. Write . is called a generalized dihedral group. In particular, if is cyclic, then is a dihedral group.

Corollary 1. Let , . Then, if , then ; and if , then .

Corollary 2. Let , . Then, if and only if either or and .

Let be an abelian group of even order and , and let . Write . is called a generalized dicyclic group. In particular, if is cyclic, then is a generalized quaternion group.

Theorem 2. Let , . Then, if , then ; and if , then .

Proof. (1)If , then is a 2-group and .(2)If , let , where are the distinct odd primes, . Then, . Case 1:  Since is an elementary abelian 2-group. It follows that , Write . Thus, , which implies that . By Lemma 3, Case 2: Let , where . Thus, is of order , andWrite , and we have thatIt follows that is of order , . By Lemma 3,

Corollary 3. Let , . Then, if , then ; if , then .

Corollary 4. Let , . Then, if and only if either or and .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was partially supported by the National Natural Science Foundation of China (11871292).