ON THE LOCAL HYPERCENTER OF A GROUP

We introduce a local hypercenter of an arbitrary group and study its basic properties. With this concept, it turns out that classical theorems of Baer, Mal’cev and McLain on locally nilpotent groups can be obtained as special cases of statements which are valid in any group. Furthermore, we investigate the connection between the local hypercenter of a group and the intersection of its maximal locally nilpotent subgroups. 2000 Mathematics Subject Classification : 20E25, 20E28, 20F14 342 José Iván Silva Ramos and Rudolf Maier The local hypercenter of a group An important canonical characteristic subgroup of an arbitrary group is its hypercenter H( ) the union of the terms of the transfinitely continued upper central series of (see [6], [7]). FollowingBaer [1], H( ) has several equivalent descriptions: It can be seen as the smallest normal subgroup of with center-free quotient and also as the largest normal subgroup of that is -hypercentral in the sense that for every such that , we have that contains a nontrivial central element of A group is hypercentral, if H( ) = This happens, if and only if every of its nontrivial homomorphic images has a nontrivial center. Clearly, nilpotent groups are hypercentral and hypercentral groups are locally nilpotent (see [6, pg. 364/365]). and indicate the classes of hypercentral and nilpotent groups, respectively, (= ) is the class of locally nilpotent groups. Since there are locally nilpotent groups with trivial (hyper-) center any restricted regular wreath product of a group of order a prime with an infinite avbelian -group is an example it is desirable to extend the concept of the hypercenter of a group in some way to a local hypercenter. A natural possibility to try this is given with the following definition, suggested to us Definition A . For any group the local hypercenter of is defined to be K( ) = n  ̄̄ ̄ H(h 1 i) 1 o This means that K( ) consists of all elements which lie in the hypercenter of any finitely generated subgroup of containing .


The local hypercenter of a group
An important canonical characteristic subgroup of an arbitrary group is its hypercenter H( ) the union of the terms of the transfinitely continued upper central series of (see [6], [7]). Following Baer [1], H( ) has several equivalent descriptions: It can be seen as the smallest normal subgroup of with center-free quotient and also as the largest normal subgroup of that is -hypercentral in the sense that for every such that , we have that contains a nontrivial central element of A group is hypercentral, if H( ) = This happens, if and only if every of its nontrivial homomorphic images has a nontrivial center. Clearly, nilpotent groups are hypercentral and hypercentral groups are locally nilpotent (see [6, pg. 364/365]). and indicate the classes of hypercentral and nilpotent groups, respectively, (= ) is the class of locally nilpotent groups. Since there are locally nilpotent groups with trivial (hyper-) center -any restricted regular wreath product of a group of order a prime with an infinite avbelian -group is an example -it is desirable to extend the concept of the hypercenter of a group in some way to a local hypercenter. A natural possibility to try this is given with the following definition, suggested to us Definition A . For any group the local hypercenter of is defined to be This means that K( ) consists of all elements which lie in the hypercenter of any finitely generated subgroup of containing .

Basic properties
The (general) hypercenter H( ) of a group, the union of the terms of the transfinitely continued upper central series of , has the following equivalent descriptions (compare [7, pg. 28, 1.39.3]): Consider the two families of normal subgroups of (by Z( ) we denote the center of ), Moreover , H( ) With these alternative descriptions of H( ), we can easily verify the following well known basic facts: be a group, a subgroup and a normal subgroup of . Then c) The radical property: d) The radical property : If K( ), then K( ) = K( ) .
In particular,

K( ) and we see that
is a hypercentral, i.e. nilpotent subgroup and we see that K( ) is locally nilpotent. It is also clear that K( ) is invariant under all automorphisms of b) For all K( ) and 1 we have c) Let For all K( ) and 1 we have by Lemma 1.1 c) and it follows that H( ). This means K( ) and therefore K( ) This shows K( be any locally nilpotent subgroup of Every finitely generated subgroup 1 of K( ) is contained in a subgroup of the form = Now H( ) and therefore 1 H( ) Since 1 is nilpotent, we see by Lemma 1.1 e) that the subgroup = H( ) 1 is hypercentral and therefore nilpotent. Thus 1 is nilpotent and so K( ) is locally nilpotent.
, the assertion follows.
The surprising utility of the concept of the local hypercenter we shall see in the following section.

On theorems of Baer-McLain and Mal'cev-McLain
In 1956, McLain [2], (see [6, 12.1.5/6, pg. 359]) published two necessary conditions for a group to be locally nilpotent : 1) If maximal subgroups exist, then they are normal in , and 2) Every chief factor of is central in .
Using our concept of the local hypercenter, we show that these properties of the locally nilpotent groups have a general meaning in every group. More exactly, we prove   This subgroup is studied with details in [4] and [5]. It can for example also be characterized as (see [5,Th. 7.1])  Obviously, H( ) if is a locally nilpotent, non-hypercent group.
Remarkable is, that K( ) also does not coincide in general with N ( ) , as is seen by the following

Proof:
Let be a Tarski

Proof:
is nilpotent and polycyclic. We prove that, if with , then contains a nontrivial central element of . Knowing this, we conclude with the maximal condition in , that there is a such that that 1 = 0 ( ) 1 ( ) ( ) = . Therefore ( ).
For the proof of Z( ) 1 we may assume = 1 We have that 1 Z( ) Let = Z( ) Case 1: If the torsion subgroup of is nontrivial, we have for some prime a finite nontrivial -subgroup with For every we have that h i N ( ) h i and it follows that h i is nilpotent. must induce a -automorphism on and therefore C ( ) is a finite -group and some nontrivial element of is in the center of Case 2: is torsion-free and is therefore the direct product of (finitely many) infinite cyclic groups. Let 1 with of minimal rank , say. For every prime , let denote the smallest subgroup of with elementary abelian -quotient. Then | | = and having infinite index  Proof: If N ( ) and is a finitely generated subgroup of containing , then N ( ) is finite, so that, by Schreier's theorem (see [6, 1.6.11]), also N ( ) is finitely generated. By 4.4 we see that N ( ) = (We observe that in A. Krassilnikov's Example (4.1), N ( ) -in fact the whole group -is a torsion group !).
To finish, we mention in this context, that Baer and B.H. Neumann characterize certain properties of groups by means of special finite coverings : A group is finite over its (hyper-) center, if and only if is covered by a finite family of abelian (hypercentral) subgroups (see [7, pg. 105/107]).
We close with the following theorem, concerning finite coverings of groups by locally nilpotent subgroups, which is analogue to Baer/Neumann's: