We study the group generated by the inertial automorphisms of an abelian group A, that is, automorphisms with the property that each subgroup H of A has finite index in the subgroup generated by H and . Clearly, contains the group of finitary automorphisms of A, which is known to be locally finite. In a previous paper, we showed that is (locally finite)-by-abelian. In this paper, we show that is also metabelian-by-(locally finite). More precisely, has a normal subgroup such that is locally finite and the derived subgroup is an abelian periodic subgroup all of whose subgroups are normal in . In the case when A is periodic, turns out to be abelian-by-(locally finite) indeed, while in the general case it is not even (locally nilpotent)-by-(locally finite). Moreover, we provide further details about the structure of .
A subgroup H of a group G is called inert if it is commensurable with its conjugates for each (see ), that is, has finite index in both H and . Groups whose subgroups are all inert have been considered in  under the name TIN-groups. The class of TIN-groups contains Tarski monsters and FC-groups (see ), that is groups in which each element has finitely many conjugates, so it is a highly complex class. However there are no infinite locally-finite simple TIN-groups, as shown in . This result has recently been extended to simple locally graded groups in . On the other hand, soluble TIN-groups have been studied in  under some finiteness conditions.
According to , if H is a subgroup of an abelian group and is endomorphism of A, then H is called -inert if H has finite index in . The interest in the concept of -inert subgroups comes from the study of the dynamical properties of a given endomorphism (see  and the bibliography therein). A subgroup H is called fully inert if it is -inert for every endomorphism of A. Fully inert subgroups of divisible abelian groups are investigated in . Then from  we recall a definition.
An endomorphism of an abelian group is called inertial endomorphism if is finite for each subgroup H. An inertial endomorphism which is bijective is called inertial automorphism.
The set of inertial endomorphisms of an abelian group A is a subring of the ring of all endomorphisms of A. We featured this subring in .
Inertial automorphisms of an abelian group may be used as a tool in the study of soluble groups whose many subgroups are inert. Recall that in  we have studied finitely generated groups in which subnormal subgroups are inert. Also recall that, on the other hand, in  it has been shown that if in a (non-abelian) group G all finitely generated subgroups H are strongly inert, that is the subgroups H and both have finite index in their join for each , then the group G is (locally finite)-by-abelian. Note that strongly inert subgroups are inert.
Recall that, when G is any group, an automorphism of G is called finitary if it acts as the identity map on a subgroup of finite index in G. Clearly finitary automorphisms of an abelian group are inertial. In  it has been shown that the group of all finitary automorphisms of a group G is both (locally finite)-by-abelian and abelian-by-(locally finite). When G is abelian, then is even locally finite ().
In this paper we study the group generated by all inertial automorphisms of an abelian group A. Clearly contains both and the subgroup of automorphisms of A leaving each subgroup invariant. In  we have shown that is locally (center-by-finite), hence (locally finite)-by-abelian. Here we consider the question of when is abelian-by-(locally finite). From the main results of this paper, it follows:
If A is a periodic abelian group, then is center-by-(locally finite).
The group is not (locally nilpotent)-by-(locally finite), provided p is an odd prime.
Recall that a group G is said to be metabelian if its derived subgroup is abelian.
If A is an abelian group, then is metabelian-by-(locally finite).
Outline of the paper.
Since some of our statements and proofs are rather technical, in Section 2 we fix notation in a rather accurate way, recall some definitions and results, and introduce concepts that will be used in the paper.
In Section 3 we highlight the role played by stability groups with respect to finitary and inertial automorphisms of A since in our investigations we shall look for abelian normal subgroups Σ of such that the automorphisms induced by via conjugation on Σ are inertial. We also describe the group in the case when A is of a type which is relevant to our purposes (Proposition 3.6) and in the case when A splits on its torsion subgroup (Proposition 3.8).
In Section 4 by Theorem 4.2 we give a rather complete description of when A is periodic. It follows Corollary A. In Example 4.4 we see that it may happen that there is no abelian normal subgroup Λ of such that .
In the final Section 6 we describe the group in detail in cases when A splits on its torsion subgroup, see Theorem 6.1 and Theorem 6.2. Thus we have Example B. Then in Example 6.3 we consider a group A for which the group is very large.
First of all, note that, since we are going to apply methods and results from  where we used the fact that inertial endomorphisms form a ring, in this paper we will regard abelian groups as right modules over their endomorphism ring and reserve the letter A for abelian groups, which are additively written.
It shall be remarked that in the previous paper  we gave a slightly more restrictive definition of inertial automorphism (and we used different notation). This is not a major problem since in the present paper we just treat the problem in greater generality, see Fact 2.12 below.
For undefined terminology, notation and basic facts we refer to  or . The letter denotes the set of all prime numbers. If , then denotes the set of prime divisors of n. If , then is called the π-component of A and is the set of primes not in π. Moreover, and , respectively, denote the torsion subgroup and the maximum divisible subgroup of the abelian group A.
If for some , we say that A is bounded. If p is a prime, by exponent of a p-group A we mean the smallest m such that , if A is bounded. Otherwise we say that A is unbounded and write . Furthermore, is the additive group of rational numbers whose denominator n is such that and , while is the ring of integers modulo n. Also, (resp. ) denotes the torsion-free rank (resp. the p-rank) of A, i.e., the cardinality of a maximal independent subset of A of elements with infinite order (resp. with p-power order). The rank of A is .
Let and . As usual, if for each and , we say that X is -invariant and denote by the restriction of to X. If we have , where for the subgroup is -invariant and , we write with respect to.
In the sequel, commutators are calculated in the holomorph group . Moreover, if is an endomorphism of the additive abelian group A and , we use the notation .
Now we recall some known facts and introduce some notions which will be used in the rest of the paper.
Power automorphisms of an abelian group.
Recall that an automorphism leaving every subgroup invariant is called a power automorphism. Clearly, the set of power automorphisms of an abelian group A is a subgroup of .
If A is an abelian p-group and (with ) is an invertible p-adic integer, then, by a slight abuse of notation, one defines the power automorphism α of A by setting , for any of order . This gives an action on A of the group of units of the ring of p-adic integers, whose image is known to be the whole . If , this action is faithful and, if , its kernel is mod . Thus we recall the following fundamental facts (see [18, Section 13.4.3 and point 9 of Exercises 13.4]).
Let A be an abelian group.
If A is a p-group and , then is isomorphic to .
If A is a p-group and , then is isomorphic to the group of units of the ring .
If A is any periodic abelian group, then is isomorphic to the cartesian product of all the groups where is the p-component of A and p ranges in .
If A is non-periodic, then , i.e. consists only of the identity map and the map .
In any case is a central subgroup of .
Invertible multiplications of an abelian group.
According to , let us consider a generalization of power automorphisms.
An automorphism is called an invertible multiplication of A if one of the following holds:
A is periodic and is a power automorphism of A,
A is non-periodic and
there exist coprime integers such that , for each .
Note that if () holds, even in the case that A is periodic, we necessarily have and . Moreover, if A is periodic, then clearly () implies (a). In any case, by abuse of notation, we will write and say that acts on A as multiplication by . Furthermore, if A is a p-group, and as defined above, we will write and say that acts on A as multiplication by α.
We warn the reader that we are using the word “multiplication” in a way different from  and that also in  the terminology is somewhat different. Moreover, sometimes we will omit writing the word “invertible”.
The invertible multiplications of an abelian group A form a central subgroup of .
If A is periodic, the statement follows from Fact 2.1. Let then A be non-periodic. It is clear that the set of invertible multiplications is a subgroup of . Let then where , for each , with coprime. Then for each we have
since . Thus . ∎
Inertial automorphisms of an abelian group.
In a lemma, we recall now some non-elementary facts that will be used in the sequel. They follow from [11, Propositions 2.2, 2.3, 3.3 and Theorem A].
Let be an automorphism of an abelian group A.
If A is torsion-free and is inertial, then is an invertible multiplication.
If , then is inertial if and only if there are a subgroup of finite index in A and an integer m such that acts as invertible multiplication by m on .
If , then is inertial if and only if
there is a torsion-free -invariant subgroup V, which is finitely generated as a -submodule, and a rational number (with m and n coprime integers) such that acts on V as the invertible multiplication by ,
is inertial on the periodic group ,
is bounded and is π -divisible, where .
If A is periodic, then inertial if and only if is inertial on each p-component of A and acts as invertible multiplication on all but finitely many of them.
If A is a p-group, then is inertial if and only if either acts as an invertible multiplication on a subgroup of finite index in A or
has finite rank and is infinite and bounded,
there is a -invariant subgroup of finite index in A such that acts as an invertible multiplication (by possibly different p-adics) on both and D.
Notice that if V is as in (2) above, then there are finitely many elements such that, in the holomorph group of A, we have that is isomorphic to the direct sum of copies of .
Invertible multiplications of an abelian group A with are inertial.
If A is periodic, the statement follows from Fact 2.1. Let then on A and where is a free subgroup of A such that is periodic. Then is contained in the π-component of . Hence V is torsion-free, as . Moreover on , thus acts as a power automorphism on . Thus, we may apply Lemma 2.4 (3) and deduce that is inertial. ∎
Now we state some facts that follow from [11, Theorem B and Corollary B].
Let A be an abelian group. Then:
the subgroup is the set of the products where and are both inertial automorphisms,
if , then is the set of inertial automorphisms,
the group is abelian.
The group of automorphisms of a non-periodic abelian group A.
Now we define a group if inertial automorphisms of A of a particular type.
Let A be a non-periodic abelian group and let p be a prime such that is p-divisible and one of the following holds:
is finite and is bounded.
Then there is a unique subgroup C such that and the automorphism (with respect to this decomposition) is an inertial automorphism of A.
The existence of C follows from the fact that is bounded (see ). For the uniqueness, if is in the same conditions as C, we have that the group is trivial, as it is both bounded and p-divisible. Since C is a fully invariant subgroup of A, each is inertial by Lemma 2.4. ∎
If is the set of primes p as in Lemma 2.7 above, then we define , where 1 and -1 denote the maps and resp.
If A is a non-periodic abelian group, then is isomorphic to the multiplicative group of units of and is a central subgroup of .
The group and finitary automorphisms of an abelian group A.
To answer the question in the Introduction, we will reduce a to a somewhat smaller group of automorphisms.
Let be the set of inertial automorphisms of the abelian group A that act as the identity map on .
Clearly is a normal subgroup of . When A is non-periodic, we have
When A is periodic, it holds
Note that in the sequel we will apply the basic fact (1) below without reference.
Let A be an abelian group. Then:
an automorphism of A is finitary if and only if the subgroup is finite,
, provided .
To prove statement (1), consider the endomorphism of A and that . To prove (2) note that if , then by (1) we have . So that acts trivially on . Finally, if , then each acts as multiplication by on a subgroup with finite index in A by Lemma 2.4 (2). By considering the endomorphism we have that is finite. Therefore acts as the multiplication by m on . If , then and . ∎
Almost-power automorphisms of an abelian group.
Recall that the group of the so-called almost-power automorphisms of A, that is, automorphisms such that every subgroup of A contains a -invariant subgroup of finite index was introduced in , where generalized soluble groups in which subnormal subgroups are normal-by-finite (or core-finite, according to the terminology of  and ) are studied.
Clearly an almost-power automorphism has the following property:
H and are commensurable for each ,
which is stronger than the property of being inertial as defined in the Introduction.
Actually, in  we called inertial an automorphism with (), while the definition of inertial that we are using in the present paper is different (and is the same as in ). However, there is no risk of misunderstandings, since applying results from  we have the following.
Let A be an abelian group. Then:
if A is periodic, then all inertial automorphisms are almost-power,
if A is non-periodic, then the group of almost-power automorphism of A is ,
if , then all inertial automorphisms have the property (),
if , then an automorphism of A with the property () is almost-power.
3 Finitary automorphisms and stability groups
We begin this section by stating some basic facts that perhaps are already known (see ).
If and X is a subgroup of A, then we denote by the stability group in of the series , that is, the set of such that and . When , then we will write .
If X is -invariant, then each acts via conjugation on the abelian -invariant subgroup of , according to the rule for each . Similarly, acts on the additive group of homomorphisms by a corresponding formula, i.e. , where and denotes the group isomorphism induced by on .
Thus, by extending the above actions (and the natural action of on any abelian group) both and are equipped with a structure of right -module, where denotes the group ring constructed in the usual way (see ).
Let A be an abelian group, and X a -invariant subgroup of A. For each , let be the well-defined homomorphism
Then the map is an isomorphism of -modules, so that for each we have
Let σ, , and . On the one hand
On the other hand we have , thus
So is a group isomorphism, the inverse being the map
where ν is the canonical homomorphism , ι is the embedding of X in A and is the identity of A. Finally, for each and , we have and is an isomorphism of -modules. ∎
By this argument we have two technical lemmas. For the first one see .
Let A be an abelian group, and . If σ stabilizes a series , where on and on , then .
Applying Fact 3.1 with and , for any , we have
Since is an isomorphism from a multiplicative to an additive group, it follows that . ∎
Our next lemma deals with the case when A splits over X and will be used several times. In such a situation, once we have fixed a direct decomposition , we have an embedding given by .
Let , , let X be a -invariant subgroup of A and let the natural isomorphism. Furthermore, let , and .
If , then and .
If , then and .
If , then .
To prove the first statement, for each , let . Then , hence . Moreover is normal in , as X is -invariant, and , as it centralizes both X and K. If , then and hence (1) is proved.
Statement (2) can be proved similarly and then (3) follows directly. ∎
Let A be an abelian group and .
If , then the automorphisms induced by via conjugation on are finitary.
If and the quotient is free abelian, then there is which induces via conjugation on a non-finitary automorphism, provided .
Thus, we have to check that the set is finite. For each σ, we have that has finite order, say n. On the other hand, and is finite since .
(2) If , where K is free abelian on the infinite -basis , take . Let such that and . For each i define by the rule and if . Then there are infinitely many , as for each . ∎
If , then .
Despite the above example, we will see that for some relevant characteristic subgroups X of A, we have , provided that one of the following holds:
has finite rank and X is bounded, as in Theorem 6.1,
is finitely generated and X is periodic, as in Theorem 6.2.
Let A be an abelian p-group such that has finite rank and is bounded. Then is a bounded abelian p-group and there is a subgroup such that
where the automorphisms induced by Φ via conjugation on Σ are finitary and this action is faithful.
First note that if , then is finite, since it is both of finite rank and bounded. Hence, we have . Consider a decomposition and apply Lemma 3.3, with and . Put . Then , as claimed.
Let . We have to show that set is finite. Since , as in Proposition 3.4, we have
Thus, we have to count how many homomorphisms there are. On the one hand, contains which has finite index in . On the other hand, the image of each is contained in the finite subgroup , where is a bound for . Therefore, there are only finitely many , once is fixed.
Let us check that the action is faithful. Let and let with maximal order and . Then we have for some and we can write with . If , then there is such that and where has the same order as b. Thus, by Fact 3.1, , while . Therefore, we have . Similarly, if , then there is such that and of order p. Then we have , while and again . ∎
We now see that, in Proposition 3.6 the picture may be rather complicated. Recall that the FC-center of a group is the set of elements with finitely many conjugates or – equivalently – whose centralizer has finite index.
There is an abelian p-group A such that has finite rank, is bounded and is not contained in the FC-center of .
Write , where and is infinite and homogeneous. Fix such that , where d is an element of D of order p, and on . For each i consider switching and acting trivially on . Then
Hence, and for each . ∎
We now use a similar argument in the case when .
Let A be an abelian group with such that is finitely generated (resp. is bounded). Then is a periodic (resp. bounded) abelian group and there is a subgroup such that
where induces via conjugation on Σ finitary automorphisms.
In the case when is finitely generated, this action by conjugation is faithful.
In any case, we can write , where . Recall that . Note that . In fact, if , then we have that and is an abelian group which is both finitely generated and periodic (resp. finite rank and bounded). Hence, is finite that is .
If is finitely generated, then is a periodic abelian group which is naturally isomorphic to the direct sum of r copies of T as a right -module. Therefore, the action of on Σ is faithful. ∎
In the notation of Proposition 3.8, if , then Σ is not self-centralizing in , that is, the action that induces via conjugation on Σ is not faithful.
We have and . ∎
4 The group when A is periodic
To give a detailed description of when A is a p-group, let us introduce some terminology.
If A is a p-group, by the essential exponent of A we mean the smallest e such that is finite, or if A is unbounded. In the former case, this is equivalent to saying that , where is finite, and is the sum of infinitely many cyclic groups of order . In  we called critical a p-group of type with B infinite but bounded and divisible with finite rank (see Lemma 2.4 (5)).
Critical groups will be a tool to describe when A is periodic.
Let A be an abelian p-group and .
If A is non-critical, then
where is either trivial or cyclic of order , according as A is unbounded or and ).
If is critical, let
Moreover, , where and the following hold:
is an infinite abelian p-group, and ,
, where and Ψ induces via conjugation on Σ inertial automorphisms and this action is faithful,
, each acts via conjugation on Σ as multiplication by n and has order ,
and Φ induces via conjugation on Σ finitary automorphisms.
(1) If A is non-critical, then, according to Lemma 2.4 (5), there exist a p-adic α and a subgroup of finite index in A such that . Thus, acts on as the identity map, that is, . Hence,
Further, if the multiplication by the p-adic number () is in , then it is trivial on a subgroup B of finite index in A. Therefore, if , then and . Otherwise, and then (see Fact 2.1 and the definition of action given before). Thus, there are at most choices for such a β. On the other hand, the invertible multiplication on A by a p-adic number is a finitary automorphisms since it acts trivially on .
(2) Let be critical. By Lemma 2.4 (5) there exists an invertible p-adic α such that . Thus, we get . Clearly, we have , so that .
Again by Lemma 2.4 (5), acts by multiplication by an integer n on a subgroup of finite index in where . Therefore, if with respect to , we have . Hence, .
It is routine to verify that (i) holds, since . By Proposition 3.6, (iv) holds as well. By Lemma 3.3 (with , and so ), we have as stated in (2). Then, applying part (1) of the statement to B, we have and as stated in (ii). Moreover, has order .
By Lemma 3.2, we have that Δ acts on Σ as in (iii). Thus, the whole of acts via conjugation on Σ inducing inertial automorphisms and (ii) holds.
It remains to show that Ψ acts faithfully on Σ. Let with and . On one hand, acts via conjugation on Σ as the multiplication by n by (iii). On the other hand, is finitary on Σ by (iv). Since by (i), then multiplication by n is finitary on B. Thus, we have by Proposition 3.6. ∎
In next statement we regard as naturally embedded in .
Let A be a periodic abelian group. Then there is a subgroup Δ of which is a direct product of finite abelian groups and such that
where Δ is trivial if A contains no non-trivial divisible subgroups.
Moreover, there are a set π of primes and subgroups Σ, Ψ of such that Σ is an abelian -group with bounded primary components and
where the automorphisms induced by Ψ via conjugation on Σ are inertial and this action is faithful.
Otherwise, for each , there are subgroups corresponding to in Proposition 4.1 such that and . Now it is routine to verify that the statement follows by setting , , , and recalling that as . ∎
Proof of Corollary A.
If is a subgroup of finite index in a bounded abelian group B, then there are subgroups and such that is finite, and
Clearly there is a finite subgroup F such that . Since is separable and is finite, then there is a finite subgroup such that for some . Fix and . On the one hand
On the other hand, by Dedekind’s law,
If A is a critical p-group (with ) and is such that , then Λ is neither finite nor locally nilpotent.
We use the same notation as in Proposition 4.1. Let be a primitive root of and consider with respect to . Since , we can assume that and . Hence, on some subgroup of finite index in B. By Lemma 4.3, with and finite. Put and note that with respect to .
It is sufficient to show that is infinite and not locally nilpotent, where is the group of (inertial) automorphisms of A of type with respect to , with . Thus, we may assume and . Then multiplication by n is in Λ and .
We claim that . In fact, by Proposition 4.1 we have that the group acts faithfully by multiplication on the infinite abelian p-group Σ of exponent and then and , as claimed. Thus, is not locally nilpotent, since the action of on Σ is fixed-point-free. ∎
Remark that, in Theorem 4.2, when we consider the action of the above Ψ on the p-component of the group Σ, we are concerned with subgroups of , where is a bounded abelian p-group and is finite abelian.
5 The group when A is non-periodic
Let us state now our main results in the non-periodic case. Recall that metabelian groups G in which each subgroup of is normal in G are called KI-groups and have been studied in a series of papers (see  and the references therein).
Let A be a non-periodic abelian group. Then there is a central subgroup of , which is isomorphic to a multiplicative group of rational numbers, such that
Moreover, there is a normal subgroup of such that:
is locally finite,
is a KI-group with periodic derived subgroup.
In particular, we have that if A is torsion-free, then is abelian, as . Further, we will prove that in the statement of Theorem 5.1 one may take to be the subgroup of consisting of inertial automorphisms acting by multiplication on . Unfortunately this subgroup need not be locally nilpotent, as in Example B.
We first consider the case when . Let . By Fact 2.6 (1), we have with , inertial. Further, by Lemma 2.4 (2), there is a subgroup with finite index in A such that we have ( coprime, prime, ). Also, and on as well. If , then . If , put . Otherwise, since is invertible, we have that . Then, for each , the -component of A is finite and is -divisible. Consider then
In both cases, on hence . Thus,
Moreover, (i) and (ii) are true with , since is locally finite.
Clearly on . Thus, acts trivially on and
Let be the preimage of under the canonical homomorphism . Now statement (i) holds, since is locally finite by Theorem 4.2. To check (ii), consider that the derived subgroup of stabilizes the series and therefore is abelian. Moreover, by Fact 2.6 (3), the subgroup consists of finitary automorphisms. Thus, is torsion and (ii) holds by Lemma 3.2. ∎
Proof of Corollary C.
Apply Theorem 5.1 and note that is metabelian where is isomorphic to a quotient of . ∎
When is not finitely generated, it may happen that A has very few inertial automorphisms.
Let A be a π-divisible non-periodic abelian group, where π is a set of primes. If is a π-group, then .
If , then . Moreover, if , then is a finite π-group. Then is such. Hence, and .
If , by Lemma 2.4 (3) we have on some free abelian subgroup such that is periodic. Moreover, the π-component of is divisible. Then, by Lemma 2.4, parts (4) and (5), we have that is a multiplication on . Furthermore, the group is π-divisible and has non-trivial p-component for each , since is free abelian. Thus, from on it follows that on . Hence, stabilizes the series . However . Then on B. Therefore, induces a homomorphism which is necessarily 0 since is a -group. Thus, on the whole group A. ∎
If , then .
6 The group when A splits on
The next two theorems consider cases in which A splits on its torsion subgroup T.
Let A be an abelian group and . If and T is bounded, then is a bounded abelian group and there is a subgroup of such that and
where induces via conjugation on Σ inertial automorphisms.
We can write , where . Note that the group is a periodic abelian group which is bounded as T.
By Proposition 4.1, we have
Hence, we have , where acts by conjugation on Σ by means of finitary automorphisms, by Proposition 3.8, and acts via conjugation on Σ by means of multiplications, by Lemma 3.2. Therefore, the whole induces by conjugation on Σ inertial automorphisms. ∎
Let A be a non-periodic abelian group and . If is finitely generated, then is a periodic abelian group and there is a subgroup of such that and
where induces via conjugation on Σ inertial automorphisms and this action is faithful.
If in addition T is unbounded, then is not nilpotent-by-(locally finite). Further, if is unbounded, then is not even (locally nilpotent)-by-(locally finite).
As in the proof of Theorem 6.1, we can write , where K is finitely generated. The group is a periodic abelian group which is isomorphic to the direct sum of copies of T as a right -module.
Let us investigate now the action of via conjugation on Σ. Assume first that T is a p-group. Let . By Proposition 4.1, we have , where and either or T is a critical p-group and induces multiplications on both and . Recall that Σ is -isomorphic to . In the former case, that is if , then acts via conjugation on Σ as a power automorphism (that is a multiplication). In the latter case, Σ is critical as well and induces invertible multiplications on both and . Thus, acts via conjugation on Σ as an inertial automorphism of Σ, by Lemma 2.4 (5). In both cases, by Proposition 3.8, acts via conjugation on Σ as a finitary automorphism. Hence, acts via conjugation on Σ as an inertial automorphism.
In the general case, when T is any periodic group and , then (with respect to ) acts via conjugation as an inertial automorphism on all primary components of Σ, by what we have seen above and the fact that . Similarly, since acts as a multiplication on all but finitely many primary components of A, it acts the same way on all but finitely many . Thus, is inertial on Σ by Lemma 2.4 (4).
It is clear that the action via conjugation of on Σ is faithful as the standard action of on T is such.
To prove the last part of the statement, note that in the case when T is unbounded, there exists a non-periodic multiplication α of T. Note that the automorphism (with respect to ) belongs to . If, by way of contradiction, is nilpotent-by-(locally finite), then there is such that is nilpotent, so there is such that , and hence . This is a contradiction, since Σ is unbounded as T is.
Finally, if the group is unbounded, then is unbounded as well. Let α be a non-periodic multiplication of . Then with respect to the group acts as non-periodic multiplication (by α) of acting fixed-point-free on a primary component. Thus, μ (and any non-trivial power of μ as well) does not belong to the locally nilpotent radical R of . Therefore, is not locally finite. ∎
Finally, we note that, despite the above propositions, in the general case the group may be large.
There exists an abelian group A with and for each prime p such that , , where with the property that and is a divisible torsion-free abelian group with cardinality .
Moreover, any element of induces a finitary automorphism on .
As in [11, Proposition A], we consider the group , where , , and , have order p, , respectively, and p ranges over all primes. Consider the (aperiodic) element and . We have that for each prime p there is an element such that . Let and . Then
since has torsion free rank 1 and has p-height 1 for each p. Thus, and the p-component of is generated by and has order , since .
Then and , hence . Moreover, , where is infinite cyclic and has order p. Also, and .
We claim that if induces on T a finitary automorphism, then . In fact, is finite, so it is a π-component of A for some finite π. Thus, , where with respect to and clearly .
Finally, we prove the last part of the statement, from which it follows that . To this end, let and . Since , there exists a non-zero integer n such that . We prove that , which is finite. For any prime p, on the one hand, is a p-element modulo , hence , that implies
On the other hand,
Hence, if , then . ∎
Dedicated to Martin L. Newell
We thank the referee for her/his useful comments.
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