# A group of generalized finitary automorphisms of an abelian group

Ulderico Dardano and Silvana Rinauro
From the journal Journal of Group Theory

# Abstract

We study the group IAut ( A ) generated by the inertial automorphisms of an abelian group A, that is, automorphisms γ with the property that each subgroup H of A has finite index in the subgroup generated by H and H γ . Clearly, IAut ( A ) contains the group FAut ( A ) of finitary automorphisms of A, which is known to be locally finite. In a previous paper, we showed that IAut ( A ) is (locally finite)-by-abelian. In this paper, we show that IAut ( A ) is also metabelian-by-(locally finite). More precisely, IAut ( A ) has a normal subgroup Γ such that IAut ( A ) / Γ is locally finite and the derived subgroup Γ is an abelian periodic subgroup all of whose subgroups are normal in Γ . In the case when A is periodic, IAut ( A ) 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 IAut ( A ) .

## 1 Introduction

A subgroup H of a group G is called inert if it is commensurable with its conjugates H g := g - 1 H g for each g G (see [1]), that is, H H g has finite index in both H and H g . Groups whose subgroups are all inert have been considered in [2] under the name TIN-groups. The class of TIN-groups contains Tarski monsters and FC-groups (see [19]), 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 [2]. This result has recently been extended to simple locally graded groups in [15]. On the other hand, soluble TIN-groups have been studied in [19] under some finiteness conditions.

According to [13], if H is a subgroup of an abelian group ( A , + ) and γ is endomorphism of A, then H is called γ -inert if H has finite index in H + H γ . The interest in the concept of γ -inert subgroups comes from the study of the dynamical properties of a given endomorphism (see [14] and the bibliography therein). A subgroup H is called fully inert if it is γ -inert for every endomorphism γ of A. Fully inert subgroups of divisible abelian groups are investigated in [13]. Then from [11] we recall a definition.

Definition

An endomorphism γ of an abelian group ( A , + ) is called inertial endomorphism if ( H + H γ ) / H 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 [9].

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 [10] we have studied finitely generated groups in which subnormal subgroups are inert. Also recall that, on the other hand, in [12] 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 H g both have finite index in their join H , H g for each g G , 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 [3] it has been shown that the group FAut ( G ) of all finitary automorphisms of a group G is both (locally finite)-by-abelian and abelian-by-(locally finite). When G is abelian, then FAut ( G ) is even locally finite ([21]).

In this paper we study the group IAut ( A ) generated by all inertial automorphisms of an abelian group A. Clearly IAut ( A ) contains both FAut ( A ) and the subgroup PAut ( A ) of automorphisms of A leaving each subgroup invariant. In [11] we have shown that IAut ( A ) is locally (center-by-finite), hence (locally finite)-by-abelian. Here we consider the question of when IAut ( A ) is abelian-by-(locally finite). From the main results of this paper, it follows:

Corollary A

If A is a periodic abelian group, then IAut ( A ) is center-by-(locally finite).

Example B

The group IAut ( ( p ) ) 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 G is abelian.

Corollary C

If A is an abelian group, then IAut ( A ) 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 IAut ( A ) such that the automorphisms induced by IAut ( A ) via conjugation on Σ are inertial. We also describe the group FAut ( A ) 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 IAut ( A ) 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 IAut ( A ) such that IAut ( A ) = Λ FAut ( A ) .

In Section 5 by Theorem 5.1 we treat the case in which A is non-periodic. It follows Corollary C. In Example 5.2 we see that | IAut ( ( p ) ( p ) ) | = 2 .

In the final Section 6 we describe the group IAut ( A ) 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 IAut ( A ) is very large.

## 2 Preliminaries

### Notation.

First of all, note that, since we are going to apply methods and results from [11] 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 [8] 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 [17] or [18]. The letter denotes the set of all prime numbers. If n , then π ( n ) denotes the set of prime divisors of n. If π , then A π := A p | p π is called the π-component of A and π is the set of primes not in π. Moreover, T ( A ) and D ( A ) , respectively, denote the torsion subgroup and the maximum divisible subgroup of the abelian group A.

If n A = 0 for some 0 n , we say that A is bounded. If p is a prime, by exponent m = exp ( A ) of a p-group A we mean the smallest m such that p m A = 0 , if A is bounded. Otherwise we say that A is unbounded and write exp ( A ) = . Furthermore, ( π ) is the additive group of rational numbers whose denominator n is such that π ( n ) π and ( p ) := ( p ) / , while ( n ) is the ring of integers modulo n. Also, r 0 ( A ) (resp. r p ( A ) ) 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 r 0 ( A ) + sup p r p ( A ) .

Let Γ Aut ( A ) and X A . As usual, if a γ X for each a X and γ Γ , we say that X is Γ -invariant and denote by γ | X the restriction of γ to X. If we have A = A 1 A 2 , where for i = 1 , 2 the subgroup A i is γ -invariant and γ | A i = γ i Aut ( A i ) , we write γ = γ 1 γ 2 with respect to A = A 1 A 2 .

In the sequel, commutators are calculated in the holomorph group A Aut ( A ) . Moreover, if φ is an endomorphism of the additive abelian group A and a A , we use the notation [ a , φ ] := a φ - a = a ( φ - 1 ) .

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 PAut ( A ) of power automorphisms of an abelian group A is a subgroup of Aut ( A ) .

If A is an abelian p-group and α = i = 0 α i p i (with 0 α i < p ) is an invertible p-adic integer, then, by a slight abuse of notation, one defines the power automorphism α of A by setting a α := ( i = 0 k - 1 α i p i ) a , for any a A of order p k . This gives an action on A of the group 𝒰 p of units of the ring of p-adic integers, whose image is known to be the whole PAut ( A ) . If exp ( A ) = , this action is faithful and, if exp ( A ) = : e < , its kernel is { α 𝒰 p α 1 mod p e } . Thus we recall the following fundamental facts (see [18, Section 13.4.3 and point 9 of Exercises 13.4]).

Fact 2.1

Let A be an abelian group.

1. (1)

If A is a p-group and exp ( A ) = , then PAut ( A ) is isomorphic to 𝒰 p .

2. (2)

If A is a p-group and e := exp ( A ) < , then PAut ( A ) is isomorphic to the group of units of the ring ( p e ) .

3. (3)

If A is any periodic abelian group, then PAut ( A ) is isomorphic to the cartesian product of all the groups PAut ( A p ) where A p is the p-component of A and p ranges in .

4. (4)

If A is non-periodic, then PAut ( A ) = { ± 1 } , i.e. PAut ( A ) consists only of the identity map and the map x - x .

5. (5)

In any case PAut ( A ) is a central subgroup of Aut ( A ) .

### Invertible multiplications of an abelian group.

According to [11], let us consider a generalization of power automorphisms.

Definition 2.2

An automorphism γ is called an invertible multiplication of A if one of the following holds:

1. (a)

A is periodic and γ is a power automorphism of A,

2. (b)

A is non-periodic and

1. ($*$)

there exist coprime integers m , n such that ( n a ) γ = m a , for each a A .

Note that if ( * ) holds, even in the case that A is periodic, we necessarily have m n A = A and A π ( m n ) = 0 . Moreover, if A is periodic, then clearly ( * ) implies (a). In any case, by abuse of notation, we will write γ = m / n and say that γ acts on A as multiplication by m / n . Furthermore, if A is a p-group, α 𝒰 p and a γ = a α as defined above, we will write γ = α and say that γ acts on A as multiplication by α.

We warn the reader that we are using the word “multiplication” in a way different from [17] and that also in [8] the terminology is somewhat different. Moreover, sometimes we will omit writing the word “invertible”.

Proposition 2.3

The invertible multiplications of an abelian group A form a central subgroup of Aut ( A ) .

### Proof.

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 Aut ( A ) . Let then γ , δ Aut ( A ) where ( n a ) γ = m a , for each a A , with m , n coprime. Then for each x A we have

n ( x δ γ - x γ δ ) = n ( x δ ) γ - ( n x γ ) δ = m ( x δ ) - ( m x ) δ = 0 .

It follows

x δ γ = x γ δ ,

since A π ( n m ) = 0 . Thus δ γ = γ δ . ∎

### 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].

Lemma 2.4

Let γ be an automorphism of an abelian group A.

1. (1)

If A is torsion-free and γ is inertial, then γ is an invertible multiplication.

2. (2)

If r 0 ( A ) = , then γ is inertial if and only if there are a subgroup A 0 of finite index in A and an integer m such that γ acts as invertible multiplication by m on A 0 .

3. (3)

If 0 < r 0 ( A ) < , then γ is inertial if and only if

1. (a)

there is a torsion-free γ -invariant subgroup V, which is finitely generated as a γ -submodule, and a rational number m / n (with m and n coprime integers) such that γ acts on V as the invertible multiplication by m / n ,

2. (b)

γ is inertial on the periodic group A / V ,

3. (c)

A π is bounded and A / A π is π -divisible, where π := π ( m n ) .

4. (4)

If A is periodic, then γ inertial if and only if γ is inertial on each p-component A p of A and acts as invertible multiplication on all but finitely many of them.

5. (5)

If A is a p-group, then γ is inertial if and only if either γ acts as an invertible multiplication on a subgroup A 0 of finite index in A or

1. (a)

D := D ( A ) 0 has finite rank and A / D is infinite and bounded,

2. (b)

there is a γ -invariant subgroup A 1 of finite index in A such that γ acts as an invertible multiplication (by possibly different p-adics) on both A 1 / D and D.

Notice that if V is as in (2) above, then there are finitely many elements a i A such that, in the holomorph group of A, we have that V = a 1 , , a n Γ is isomorphic to the direct sum of r 0 ( A ) copies of ( π ) .

Corollary 2.5

Invertible multiplications of an abelian group A with r 0 ( A ) < are inertial.

### Proof.

If A is periodic, the statement follows from Fact 2.1. Let then γ = m / n on A and V = F γ where F 0 is a free subgroup of A such that A / F is periodic. Then V / F is contained in the π-component of A / F . Hence V is torsion-free, as A π ( m n ) = 0 . Moreover γ = m / n on A / V , thus γ acts as a power automorphism on A / V . Thus, we may apply Lemma 2.4 (3) and deduce that γ is inertial. ∎

Now we state some facts that follow from [11, Theorem B and Corollary B].

Fact 2.6

Let A be an abelian group. Then:

1. (1)

the subgroup IAut ( A ) is the set of the products γ 1 γ 2 - 1 where γ 1 and γ 2 are both inertial automorphisms,

2. (2)

if r 0 ( A ) < , then IAut ( A ) is the set of inertial automorphisms,

3. (3)

the group IAut ( A ) / FAut ( A ) is abelian.

### The group Q ⁢ ( A ) of automorphisms of a non-periodic abelian group A.

Now we define a group Q ( A ) if inertial automorphisms of A of a particular type.

Lemma 2.7

Let A be a non-periodic abelian group and let p be a prime such that A / A p is p-divisible and one of the following holds:

1. (i)

A p is finite,

2. (ii)

r 0 ( A ) is finite and A p is bounded.

Then there is a unique subgroup C such that A = A p C and the automorphism γ ( p ) := 1 p (with respect to this decomposition) is an inertial automorphism of A.

### Proof.

The existence of C follows from the fact that A p is bounded (see [17]). For the uniqueness, if C 1 is in the same conditions as C, we have that the group C / C C 1 is trivial, as it is both bounded and p-divisible. Since C is a fully invariant subgroup of A, each γ ( p ) is inertial by Lemma 2.4. ∎

Definition 2.8

If π * ( A ) is the set of primes p as in Lemma 2.7 above, then we define Q ( A ) := γ ( p ) p π * ( A ) × { ± 1 } , where 1 and -1 denote the maps id A and - id A resp.

Proposition 2.9

If A is a non-periodic abelian group, then Q ( A ) is isomorphic to the multiplicative group of units of Q ( π * ( A ) ) and is a central subgroup of IAut ( A ) .

### Proof.

Apply Lemma 2.7, Proposition 2.3 and note that the rule - 1 - id A and p γ ( p ) (for each p) defines the wished isomorphism. ∎

### The group IAut 1 ⁡ ( A ) 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.

Definition 2.10

Let IAut 1 ( A ) be the set of inertial automorphisms of the abelian group A that act as the identity map on A / T ( A ) .

Clearly IAut 1 ( A ) is a normal subgroup of IAut 1 ( A ) . When A is non-periodic, we have

IAut 1 ( A ) PAut ( A ) = 1 .

When A is periodic, it holds

IAut 1 ( A ) = IAut ( A ) .

Note that in the sequel we will apply the basic fact (1) below without reference.

Proposition 2.11

Let A be an abelian group. Then:

1. (1)

an automorphism γ of A is finitary if and only if the subgroup A ( γ - 1 ) is finite,

2. (2)

FAut ( A ) IAut 1 ( A ) ,

3. (3)

FAut ( A ) = IAut 1 ( A ) , provided r 0 ( A ) = .

### Proof.

To prove statement (1), consider the endomorphism γ - 1 of A and that A ( γ - 1 ) ker ( γ - 1 ) . To prove (2) note that if γ FAut ( A ) , then by (1) we have A ( γ - 1 ) T ( A ) . So that γ acts trivially on A / T ( A ) . Finally, if r 0 ( A ) = , then each γ IAut ( G ) acts as multiplication by m on a subgroup with finite index in A by Lemma 2.4 (2). By considering the endomorphism γ - m we have that A ( γ - m ) A / ker ( φ - m ) is finite. Therefore γ acts as the multiplication by m on A / T ( A ) . If γ IAut 1 ( A ) , then m = 1 and γ FAut ( A ) . ∎

### Almost-power automorphisms of an abelian group.

Recall that the group of the so-called almost-power automorphisms of A, that is, automorphisms γ such that every subgroup of A contains a γ -invariant subgroup of finite index was introduced in [16], where generalized soluble groups in which subnormal subgroups are normal-by-finite (or core-finite, according to the terminology of [4] and [6]) are studied.

Clearly an almost-power automorphism γ has the following property:

1. (${\dagger}$)

H and H γ are commensurable for each H A ,

which is stronger than the property of being inertial as defined in the Introduction.

Actually, in [8] we called inertial an automorphism γ with ( ), while the definition of inertial that we are using in the present paper is different (and is the same as in [11]). However, there is no risk of misunderstandings, since applying results from [8] we have the following.

Fact 2.12

Let A be an abelian group. Then:

1. (1)

if A is periodic, then all inertial automorphisms are almost-power,

2. (2)

if A is non-periodic, then the group of almost-power automorphism of A is IAut 1 ( A ) × { ± 1 } ,

3. (3)

if r 0 ( A ) < , then all inertial automorphisms γ have the property ( ),

4. (4)

if r 0 ( A ) = , then an automorphism γ 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 [5]).

If Γ Aut ( A ) and X is a subgroup of A, then we denote by St Γ ( A , X ) the stability group in Γ of the series A X 0 , that is, the set of γ Γ such that X [ A , γ ] := A ( γ - 1 ) and [ X , γ ] = 0 . When Γ = Aut ( A ) , then we will write St ( A , X ) := St Aut ( G ) ( A , X ) .

If X is Γ -invariant, then each γ Γ acts via conjugation on the abelian Γ -invariant subgroup Σ := St ( A , X ) of Γ , according to the rule σ γ - 1 σ γ = : σ γ for each σ Σ . Similarly, γ acts on the additive group Hom ( A / X , X ) of homomorphisms A / X X by a corresponding formula, i.e. φ γ | A / X - 1 φ γ | X , where φ Hom ( A / X , X ) and γ | A / X denotes the group isomorphism induced by γ on A / X .

Thus, by extending the above actions (and the natural action of on any abelian group) both St ( A , X ) and Hom ( A / X , X ) are equipped with a structure of right Γ -module, where Γ denotes the group ring constructed in the usual way (see [18]).

Fact 3.1

Let A be an abelian group, Γ Aut ( A ) and X a Γ -invariant subgroup of A. For each σ S t Γ ( A , X ) , let σ be the well-defined homomorphism

a ¯ A / X a σ - a X .

Then the map : σ St ( A , X ) σ Hom ( A / X , X ) is an isomorphism of Γ -modules, so that for each γ Aut ( A ) we have

( σ γ ) = γ | A / X - 1 ( σ ) γ | X .

## Proof.

Let σ, τ St ( A , X ) , and a ¯ A / X . On the one hand

a ¯ ( σ + τ ) = ( a σ - a ) + ( a τ - a ) .

On the other hand we have ( a σ - a ) τ - ( a σ - a ) = 0 , thus

a ¯ ( σ τ ) = a σ τ - a = a τ + a σ - 2 a .

So is a group isomorphism, the inverse being the map

φ Hom ( A / X , X ) ν φ ι + id St ( A , X ) ,

where ν is the canonical homomorphism A A / X , ι is the embedding of X in A and id is the identity of A. Finally, for each γ Aut ( A ) and a ¯ A / X , we have a ¯ ( σ γ ) = a σ γ - a = a γ - 1 ( σ - 1 ) γ = a ¯ ( γ | A / X - 1 ( σ ) γ | X ) and is an isomorphism of Γ -modules. ∎

By this argument we have two technical lemmas. For the first one see [7].

Lemma 3.2

Let A be an abelian group, σ , γ Aut ( A ) and m 1 , m 2 Z . If σ stabilizes a series 0 A 1 A , where γ = m 1 on A 1 and γ - 1 = m 2 on A / A 1 , then σ γ = σ m 1 m 2 .

## Proof.

Applying Fact 3.1 with X := A 1 and Γ = γ , for any a ¯ A / X , we have

a ¯ ( σ γ ) = a ¯ γ | A / X - 1 ( σ ) γ | X = ( m 2 a ¯ ) ( σ ) γ | X = m 1 ( ( m 2 a ¯ ) σ ) = a ¯ ( m 1 m 2 ( σ ) ) .

Since is an isomorphism from a multiplicative to an additive group, it follows that σ γ = σ m 1 m 2 . ∎

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 A = X K , we have an embedding Aut ( K ) Aut ( A ) given by γ 1 γ .

Lemma 3.3

Let A = X K , Γ Aut ( A ) , let X be a Γ -invariant subgroup of A and let ζ : A / X K the natural isomorphism. Furthermore, let Σ := St Γ ( A , X ) , Γ 1 := { γ | X 1 γ Γ } and Γ 2 := { 1 ζ - 1 γ | A / X ζ γ Γ } .

1. (1)

If Γ 1 Γ , then Γ = C Γ ( X ) Γ 1 and C Γ ( A / X ) = Σ Γ 1 .

2. (2)

If Γ 2 Γ , then Γ = C Γ ( A / X ) Γ 2 and C Γ ( X ) = Σ Γ 2 .

3. (3)

If Γ 1 Γ 2 Γ , then Γ = Σ ( Γ 1 × Γ 2 ) .

## Proof.

To prove the first statement, for each γ Γ , let γ ¯ := γ | X 1 Γ 1 . Then δ := γ γ ¯ - 1 C Γ ( X ) , hence Γ = C Γ ( X ) Γ 1 . Moreover C Γ ( X ) is normal in Γ , as X is Γ -invariant, and C Γ ( X ) Γ 1 = 1 , as it centralizes both X and K. If γ C Γ ( A / X ) , then δ Σ and hence (1) is proved.

Statement (2) can be proved similarly and then (3) follows directly. ∎

Proposition 3.4

Let A be an abelian group and T := T ( A ) .

1. (1)

If r 0 ( A ) < , then the automorphisms induced by FAut ( A ) via conjugation on St ( A , T ) are finitary.

2. (2)

If r 0 ( A ) = and the quotient A / T is free abelian, then there is γ FAut ( A ) which induces via conjugation on St ( A , T ) a non-finitary automorphism, provided FAut ( T ) 1 .

## Proof.

(1) Denote A ¯ = A / T and fix γ FAut ( A ) . Then γ | A / T = 1 by Proposition 2.11. So by Fact 3.1 with X := T , for each σ St ( A , T ) we have

[ σ , γ ] = ( σ - 1 σ γ ) = - σ + σ γ = - σ + σ γ | T = σ ( γ | T - 1 ) = : φ σ .

Thus, we have to check that the set { φ σ σ St ( A , T ) } is finite. For each σ, we have that i m ( φ σ ) i m ( γ - 1 ) has finite order, say n. On the other hand, ker ( φ σ ) n A ¯ and A ¯ / n A ¯ is finite since r 0 ( A ¯ ) < .

(2) If A = T K , where K is free abelian on the infinite -basis { a i } , take γ 0 FAut ( T ) { 1 } . Let t T such that t γ 0 t and γ := γ 0 1 . For each i define σ i St ( A , T ) by the rule a i ( σ i - 1 ) := t and a j ( σ i - 1 ) := 0 if j i . Then there are infinitely many [ σ i , γ ] , as a i ker ( [ σ i , γ ] ) a j for each i j . ∎

Example 3.5

If A = ( p ) ( p ) , then St ( A , T ( A ) ) FAut ( A ) = 1 .

## Proof.

By Fact 3.1 we have that St ( A , T ( A ) ) Hom ( ( p ) , ( p ) ) is infinite, while by using Proposition 2.11 (1) it is easy to check that FAut ( A ) = 1 . (see also Example 5.3). ∎

Despite the above example, we will see that for some relevant characteristic subgroups X of A, we have St ( A , X ) FAut ( A ) , provided that one of the following holds:

• A / X is bounded and X has finite rank, as in Propositions 3.6 and 4.1 (2),

• A / X has finite rank and X is bounded, as in Theorem 6.1,

• A / X is finitely generated and X is periodic, as in Theorem 6.2.

Proposition 3.6

Let A be an abelian p-group such that D := D ( A ) has finite rank and A / D is bounded. Then Σ := St ( A , D ) is a bounded abelian p-group and there is a subgroup Φ FAut ( A / D ) such that

FAut ( A ) = Σ Φ ,

where the automorphisms induced by Φ via conjugation on Σ are finitary and this action is faithful.

## Proof.

First note that if σ Σ , then [ A , σ ] = A ( σ - 1 ) is finite, since it is both of finite rank and bounded. Hence, we have σ FAut ( A ) . Consider a decomposition A = D B and apply Lemma 3.3, with X = D and Γ = FAut ( A ) = C Γ ( X ) . Put Φ := Γ 2 . Then FAut ( A ) = Σ Φ , as claimed.

Let γ Φ . We have to show that set { [ σ , γ ] σ Σ } is finite. Since γ | D = 1 , as in Proposition 3.4, we have

[ σ , γ ] = ( σ - 1 σ γ ) = - σ + σ γ = - σ + γ | A / D - 1 σ = ( γ | A / D - 1 - 1 ) σ = : φ σ .

Thus, we have to count how many homomorphisms φ σ there are. On the one hand, ker ( φ σ ) contains ker ( γ | A / D - 1 - 1 ) which has finite index in A / D . On the other hand, the image of each φ σ is contained in the finite subgroup D [ p m ] , where p m is a bound for A / D . Therefore, there are only finitely many φ σ , once γ is fixed.

Let us check that the action is faithful. Let 1 γ Φ and let b B with maximal order and b b γ . Then we have B = b B 0 for some B 0 B and we can write b γ = n b + b 0 with n , b 0 B 0 . If b n b , then there is σ Σ such that B 0 ( σ - 1 ) = 0 and b ( σ - 1 ) = d where d D has the same order as b. Thus, by Fact 3.1, b γ ( σ γ - 1 ) = b γ ( γ - 1 ( σ - 1 ) ) = d , while b γ ( σ - 1 ) = n d . Therefore, we have σ γ σ . Similarly, if b = n b , then there is σ Σ such that b ( σ - 1 ) = 0 and b 0 ( σ - 1 ) = d 1 of order p. Then we have b γ ( σ γ - 1 ) = 0 , while b γ ( σ - 1 ) = d 1 and again σ γ σ . ∎

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.

Example 3.7

There is an abelian p-group A such that D := D ( A ) has finite rank, A / D is bounded and Σ := St ( A , D ) is not contained in the FC-center of FAut ( A ) .

## Proof.

Write A = D B 0 , where D ( p ) and B 0 = i b i B is infinite and homogeneous. Fix σ Σ such that b 1 ( σ - 1 ) = d , where d is an element of D of order p, and σ - 1 = 0 on D j 1 b j . For each i consider γ i FAut ( A ) switching b i b 1 and acting trivially on D ( j { 1 , i } b j ) . Then

σ γ i = γ i - 1 ( σ - 1 ) + 1 .

Hence, b i σ γ i = d + b i and b j σ γ i = b j for each j i . ∎

We now use a similar argument in the case when X = T ( A ) .

Proposition 3.8

Let A be an abelian group with r 0 ( A ) < such that A / T is finitely generated (resp. T := T ( A ) is bounded). Then Σ := St ( A , T ) is a periodic (resp. bounded) abelian group and there is a subgroup Φ 1 FAut ( T ) such that

FAut ( A ) = Σ Φ 1 ,

where Φ 1 induces via conjugation on Σ finitary automorphisms.

In the case when A / T 0 is finitely generated, this action by conjugation is faithful.

## Proof.

In any case, we can write A = T K , where r := r 0 ( K ) < . Recall that Σ Hom ( A / T , T ) . Note that Σ FAut ( A ) . In fact, if σ Σ , then we have that σ - 1 Hom ( A / T , T ) and A ( σ - 1 ) is an abelian group which is both finitely generated and periodic (resp. finite rank and bounded). Hence, A ( σ - 1 ) is finite that is σ FAut ( A ) .

Clearly Φ 1 := { φ 1 φ FAut ( T ) } FAut ( T ) and Φ 1 FAut ( A ) . Then, by Lemma 3.3 (1) we have that FAut ( A ) = Σ Φ 1 . By Proposition 3.4, Φ 1 induces via conjugation on Σ finitary automorphisms.

If A / T is finitely generated, then Σ Hom ( A / T , T ) is a periodic abelian group which is naturally isomorphic to the direct sum of r copies of T as a right Aut ( A ) -module. Therefore, the action of Φ 1 on Σ is faithful. ∎

Example 3.9

In the notation of Proposition 3.8, if A = 12 ( 2 ) , then Σ is not self-centralizing in FAut ( A ) , that is, the action that Φ 1 induces via conjugation on Σ is not faithful.

## Proof.

We have FAut ( A ) / Σ 𝒰 12 and Σ 3 . ∎

## 4 The group IAut ⁡ ( A ) when A is periodic

To give a detailed description of IAut ( A ) when A is a p-group, let us introduce some terminology.

Definition

If A is a p-group, by the essential exponent e = eexp ( A ) of A we mean the smallest e such that p e A is finite, or e = if A is unbounded. In the former case, this is equivalent to saying that A = A 0 A 1 A 2 , where A 0 is finite, exp ( A 1 ) < e exp ( A 0 ) and A 2 is the sum of infinitely many cyclic groups of order p e . In [8] we called critical a p-group of type A = B D with B infinite but bounded and D 0 divisible with finite rank (see Lemma 2.4 (5)).

Critical groups will be a tool to describe IAut ( A ) when A is periodic.

Proposition 4.1

Let A be an abelian p-group and D := D ( A ) .

1. (1)

If A is non-critical, then

IAut ( A ) = PAut ( A ) FAut ( A ) ,

where PAut ( A ) FAut ( A ) is either trivial or cyclic of order p m - e , according as A is unbounded or m := exp ( A ) < and e := eexp ( A ).

2. (2)

If A = D B is critical, let

Δ := { 1 n n p } , Φ := { 1 φ 0 φ 0 FAut ( B ) } , Ψ := { 1 γ 0 γ 0 IAut ( B ) } .

Then

IAut ( A ) = PAut ( A ) × ( FAut A Δ ) .

Moreover, FAut A Δ = C IAut ( A ) ( D ) = Σ Ψ , where FAut ( A ) = Σ Φ and the following hold:

1. (i)

Σ := St ( A , D ) is an infinite abelian p-group, exp ( Σ ) = exp ( B ) = : m < and eexp ( Σ ) = eexp ( B ) = : e ,

2. (ii)

Ψ = Φ Δ IAut ( B ) , where [ Φ , Δ ] = 1 and Ψ induces via conjugation on Σ inertial automorphisms and this action is faithful,

3. (iii)

Δ PAut ( B ) 𝒰 ( ( p m ) ) , each δ n := 1 n Δ acts via conjugation on Σ as multiplication by n and FAut ( A ) Δ has order p m - e ,

4. (iv)

Φ FAut ( B ) and Φ induces via conjugation on Σ finitary automorphisms.

## Proof.

Let γ Γ := IAut ( A ) .

(1) If A is non-critical, then, according to Lemma 2.4 (5), there exist a p-adic α and a subgroup A 0 of finite index in A such that γ | A 0 = α . Thus, γ - 1 α acts on A 0 as the identity map, that is, γ - 1 α FAut ( A ) . Hence,

IAut ( A ) = PAut ( A ) FAut ( A ) .

Further, if the multiplication by the p-adic number β = i β i p i ( 0 β i < p ) is in PAut ( A ) FAut ( A ) , then it is trivial on a subgroup B of finite index in A. Therefore, if exp ( A ) = , then exp ( B ) = and β = 1 . Otherwise, exp ( B ) e and then β 1 + i = e + 1 m - 1 β i p i mod p m (see Fact 2.1 and the definition of action given before). Thus, there are at most p m - e choices for such a β. On the other hand, the invertible multiplication on A by a p-adic number β 1 mod p e is a finitary automorphisms since it acts trivially on A [ p e ] .

(2) Let A = D B be critical. By Lemma 2.4 (5) there exists an invertible p-adic α such that γ | D = α . Thus, we get γ 1 := γ α - 1 C Γ ( D ) . Clearly, we have PAut ( A ) C Γ ( D ) = 1 , so that IAut ( A ) = PAut ( A ) × C Γ ( D ) .

Again by Lemma 2.4 (5), γ 1 acts by multiplication by an integer n on a subgroup of finite index in A [ p m ] where A [ p m ] B . Therefore, if δ n := 1 n Δ with respect to A = D B , we have γ 1 δ n - 1 FAut ( A ) . Hence, C Γ ( D ) = FAut ( A ) Δ .

It is routine to verify that (i) holds, since Σ := St ( A , D ) Hom ( B , D ) . By Proposition 3.6, (iv) holds as well. By Lemma 3.3 (with X := D , K := B and so Γ 2 = Ψ ), we have C Γ ( D ) = Σ Ψ as stated in (2). Then, applying part (1) of the statement to B, we have [ Φ , Δ ] = 1 and Ψ = Φ Δ = Δ Φ as stated in (ii). Moreover, FAut ( A ) Δ has order p m - e .

By Lemma 3.2, we have that Δ acts on Σ as in (iii). Thus, the whole of Ψ = Δ Φ acts via conjugation on Σ inducing inertial automorphisms and (ii) holds.

It remains to show that Ψ acts faithfully on Σ. Let φ δ n C Ψ ( Σ ) with φ Φ and δ n := 1 n Δ . On one hand, δ n acts via conjugation on Σ as the multiplication by n by (iii). On the other hand, δ n is finitary on Σ by (iv). Since eexp ( Σ ) = eexp ( B ) by (i), then multiplication by n is finitary on B. Thus, we have δ n C Φ ( Σ ) = 1 by Proposition 3.6. ∎

In next statement we regard FAut ( A π ) as naturally embedded in FAut ( A ) .

Theorem 4.2

Let A be a periodic abelian group. Then there is a subgroup Δ of IAut ( A ) which is a direct product of finite abelian groups and such that

IAut ( A ) = PAut ( A ) FAut ( A ) Δ ,

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 IAut ( A ) such that Σ is an abelian π -group with bounded primary components and

FAut ( A ) Δ = FAut ( A π ) × ( Σ Ψ ) ,

where the automorphisms induced by Ψ via conjugation on Σ are inertial and this action is faithful.

## Proof.

From Lemma 2.4 (4) we know that the group IAut ( A ) may be identified with PAut ( A ) Dr p IAut ( A p ) . Apply Proposition 4.1 to each A p . Let π be the set of primes p for which A p is not critical. If p π , we have

IAut ( A p ) = PAut ( A p ) FAut ( A p ) .

Otherwise, for each p π , there are subgroups Δ p , Σ p , Ψ p corresponding to Δ , Σ , Ψ in Proposition 4.1 such that IAut ( A p ) = PAut ( A p ) FAut ( A p ) Δ p and FAut ( A p ) Δ p = Σ p Ψ p . Now it is routine to verify that the statement follows by setting Δ := Dr p π Δ p , Σ := Dr p π Σ p , Ψ := Dr p π Ψ p , and recalling that FAut ( A ) = Dr p FAut ( A p ) as A = p A p . ∎

## Proof of Corollary A.

In the notation of Theorem 4.2, we have that Δ is periodic abelian, PAut ( A ) is central and FAut ( A ) is locally finite as recalled in Section 2 and the Introduction, respectively. ∎

Lemma 4.3

If B 0 is a subgroup of finite index in a bounded abelian group B, then there are subgroups B 1 and B 2 such that B 2 is finite, B 1 B 0 and B = B 1 B 2 .

## Proof.

Clearly there is a finite subgroup F such that B = B 0 + F . Since B 0 is separable and B 0 F is finite, then there is a finite subgroup B 3 B 0 F such that B 0 = B 1 B 3 for some B 1 B 0 . Fix B 1 and B 2 := B 3 + F . On the one hand

B 1 + B 2 = B 1 + B 3 + F = B 0 + F = B .

On the other hand, by Dedekind’s law,

B 1 B 2 = B 1 ( B 3 + F )
= B 1 ( B 0 ( B 3 + F ) )
= B 1 ( B 3 + ( B 0 F ) )
= B 1 B 3 = 0 .

Example 4.4

If A is a critical p-group (with p 2 ) and Λ IAut ( A ) is such that C Γ ( D ) = FAut ( A ) Λ , then Λ is neither finite nor locally nilpotent.

## Proof.

We use the same notation as in Proposition 4.1. Let n be a primitive root of 1 mod p m and consider δ := 1 n Δ with respect to A = D B . Since Δ C Γ ( D ) = FAut ( A ) Λ , we can assume that φ FAut ( A ) and λ Λ . Hence, δ = λ = n on some subgroup B 0 of finite index in B. By Lemma 4.3, B = B 1 B 2 with B 1 B 0 and B 2 finite. Put A 1 := D + B 1 and note that λ | A 1 = 1 n with respect to A 1 = D B 1 .

It is sufficient to show that λ Γ 1 is infinite and not locally nilpotent, where Γ 1 is the group of (inertial) automorphisms of A of type γ 1 1 with respect to A = A 1 B 2 , with γ 1 IAut ( A 1 ) . Thus, we may assume A 1 = A and Γ := Γ 1 . Then multiplication by n is in Λ and Λ = Δ Γ .

We claim that Δ Γ = Σ Δ . In fact, by Proposition 4.1 we have that the group Δ 𝒰 ( p m ) acts faithfully by multiplication on the infinite abelian p-group Σ of exponent m and then Σ = [ Σ , Δ ] and Δ Γ = Σ Δ , as claimed. Thus, Δ Γ is not locally nilpotent, since the action of Δ 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 Σ p of the group Σ, we are concerned with subgroups of IAut ( Σ p ) = PAut ( Σ p ) FAut ( Σ p ) , where Σ p is a bounded abelian p-group and PAut ( Σ p ) is finite abelian.

## 5 The group IAut ⁡ ( A ) 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 G is normal in G are called KI-groups and have been studied in a series of papers (see [20] and the references therein).

Theorem 5.1

Let A be a non-periodic abelian group. Then there is a central subgroup Q ( A ) of IAut ( A ) , which is isomorphic to a multiplicative group of rational numbers, such that

IAut ( A ) = IAut 1 ( A ) × Q ( A ) .

Moreover, there is a normal subgroup Γ of IAut 1 ( A ) such that:

1. (i)

IAut 1 ( A ) / Γ is locally finite,

2. (ii)

Γ is a KI-group with periodic derived subgroup.

In particular, we have that if A is torsion-free, then IAut ( A ) = Q ( A ) is abelian, as IAut 1 ( A ) = 1 . Further, we will prove that in the statement of Theorem 5.1 one may take Γ to be the subgroup of IAut 1 ( A ) consisting of inertial automorphisms acting by multiplication on T ( A ) . Unfortunately this subgroup need not be locally nilpotent, as in Example B.

## Proof.

Let γ ( p ) and Q := Q ( A ) as in Lemmata 2.7, 2.9 and Definition 2.8.

We first consider the case when r 0 ( A ) = . Let γ IAut ( A ) . By Fact 2.6 (1), we have γ = γ 1 γ 2 - 1 with γ 1 , γ 2 inertial. Further, by Lemma 2.4 (2), there is a subgroup A 0 with finite index in A such that we have γ | A 0 = m / n = p 1 s 1 p t s t ( m , n coprime, p i prime, s i ). Also, IAut 1 ( A ) = FAut ( A ) and γ = m / n on A / T as well. If m / n = 1 , then γ FAut ( A ) . If m / n = - 1 , put γ 0 := - 1 Q . Otherwise, since γ is invertible, we have that m A 0 = A 0 = n A 0 . Then, for each p i π := π ( m n ) , the p i -component of A is finite and A / T is p i -divisible. Consider then

γ 0 := γ ( p 1 ) s 1 γ ( p t ) s t Q .

In both cases, γ γ 0 - 1 = 1 on A 0 / ( A 0 ) π hence γ γ 0 - 1 FAut ( A ) . Thus,

IAut ( A ) = IAut 1 ( A ) × Q ( A ) .

Moreover, (i) and (ii) are true with Γ = 1 , since IAut 1 ( A ) = FAut ( A ) is locally finite.

Let then r 0 ( A ) < and γ IAut ( A ) . By Fact 2.6 (2), γ is inertial. By Lemma 2.4 (3), we have that γ = m / n = p 1 s 1 p t s t ( m , n coprime, p i prime, s i ) on A / T . We also have that, for each p i π := π ( m n ) , the group A / T is p i -divisible and A p i is bounded. Consider

γ 0 := γ ( p 1 ) s 1 γ ( p t ) s t Q .

Clearly γ 0 = m / n on A / T . Thus, γ γ 0 - 1 acts trivially on A / T and

IAut ( A ) = IAut 1 ( A ) × Q ( A ) ,

as stated.

Let Γ be the preimage of PAut ( T ) under the canonical homomorphism IAut 1 ( A ) IAut ( T ) . Now statement (i) holds, since IAut 1 ( A ) / Γ is locally finite by Theorem 4.2. To check (ii), consider that the derived subgroup Γ of Γ stabilizes the series 0 T A and therefore is abelian. Moreover, by Fact 2.6 (3), the subgroup Γ consists of finitary automorphisms. Thus, Γ is torsion and (ii) holds by Lemma 3.2. ∎

## Proof of Corollary C.

Apply Theorem 5.1 and note that Q ( A ) Γ is metabelian where IAut ( A ) / Q ( A ) Γ is isomorphic to a quotient of IAut 1 ( A ) / Γ . ∎

When A / T is not finitely generated, it may happen that A has very few inertial automorphisms.

Proposition 5.2

Let A be a π-divisible non-periodic abelian group, where π is a set of primes. If T := T ( A ) is a π-group, then IAut 1 ( A ) = 1 .

## Proof.

If r 0 ( A ) = , then IAut 1 ( A ) = FAut ( A ) . Moreover, if γ FAut ( A ) , then A ( γ - 1 ) is a finite π-group. Then A / ker ( γ - 1 ) is such. Hence, A = ker ( γ - 1 ) and FAut ( A ) = 1 .

If r 0 ( A ) < , by Lemma 2.4 (3) we have γ = 1 on some free abelian subgroup V A such that A / V is periodic. Moreover, the π-component B / V of A / V is divisible. Then, by Lemma 2.4, parts (4) and (5), we have that γ is a multiplication on B / V . Furthermore, the group B / ( V + T ) is π-divisible and has non-trivial p-component for each p π , since ( V + T ) / T V is free abelian. Thus, from γ = 1 on B / T it follows that γ = 1 on γ = 1 . Hence, γ stabilizes the series 0 V B . However Hom ( B / V , V ) = 0 . Then γ = 1 on B. Therefore, γ - 1 induces a homomorphism A / B T which is necessarily 0 since A / B is a π -group. Thus, γ = 1 on the whole group A. ∎

From Proposition 5.2 and Lemma 2.4 (3) we have

Example 5.3

If A = ( p ) ( p ) , then IAut ( A ) = { ± 1 } .

## 6 The group IAut ⁡ ( A ) when A splits on T ⁢ ( A )

The next two theorems consider cases in which A splits on its torsion subgroup T.

Theorem 6.1

Let A be an abelian group and T := T ( A ) . If r 0 ( A ) < and T is bounded, then Σ := St ( A , T ) is a bounded abelian group and there is a subgroup Γ 1 of IAut 1 ( A ) such that Γ 1 IAut ( T ) and

IAut 1 ( A ) = Σ Γ 1 ,

where Γ 1 induces via conjugation on Σ inertial automorphisms.

## Proof.

We can write A = T K , where r := r 0 ( K ) < . Note that the group Σ Hom ( A / T , T ) is a periodic abelian group which is bounded as T.

Clearly Γ 1 := { γ 1 γ IAut ( T ) } IAut ( T ) . If γ IAut ( T ) , then γ 1 (with respect to T K ) is inertial by Lemma 2.4 (5), and so Γ 1 IAut 1 ( A ) . Thus, we may apply Lemma 3.3 with Γ := IAut 1 ( A ) . We obtain IAut 1 ( A ) = Σ Γ 1 , as claimed.

By Proposition 4.1, we have

IAut ( T ) = FAut ( T ) PAut ( T ) .

Hence, we have Γ 1 = Φ 1 Δ 1 , where Φ 1 := { φ 1 φ FAut ( T ) } FAut ( T ) acts by conjugation on Σ by means of finitary automorphisms, by Proposition 3.8, and Δ 1 := { δ 1 δ PAut ( T ) } PAut ( T ) acts via conjugation on Σ by means of multiplications, by Lemma 3.2. Therefore, the whole Γ 1 induces by conjugation on Σ inertial automorphisms. ∎

We notice that the action of Γ 1 on Σ in Theorem 6.1 need not be faithful, as already seen in Proposition 3.8.

Theorem 6.2

Let A be a non-periodic abelian group and T := T ( A ) . If A / T is finitely generated, then Σ := St ( A , T ) is a periodic abelian group and there is a subgroup Γ 1 of IAut 1 ( A ) such that Γ 1 IAut ( T ) and

IAut 1 ( A ) = Σ Γ 1

where Γ 1 induces via conjugation on Σ inertial automorphisms and this action is faithful.

If in addition T is unbounded, then IAut 1 ( A ) is not nilpotent-by-(locally finite). Further, if A 2 is unbounded, then IAut 1 ( A ) is not even (locally nilpotent)-by-(locally finite).

## Proof.

As in the proof of Theorem 6.1, we can write A = T K , where K is finitely generated. The group Σ Hom ( A / T , T ) is a periodic abelian group which is isomorphic to the direct sum r T of r := r 0 ( A ) > 0 copies of T as a right Aut ( A ) -module.

Clearly Γ 1 := { γ 1 γ IAut ( T ) } IAut ( T ) . If γ IAut ( T ) , then γ 1 (with respect to T K ) is inertial by Lemma 2.4 (5). Hence, Γ 1 IAut 1 ( A ) . Thus, we may apply Lemma 3.3 with Γ := IAut 1 ( A ) , and we obtain

IAut 1 ( A ) = Σ Γ 1 .

Let us investigate now the action of Γ 1 via conjugation on Σ. Assume first that T is a p-group. Let γ IAut ( T ) . By Proposition 4.1, we have γ = γ 0 φ , where φ FAut ( T ) and either γ 0 PAut ( T ) or T is a critical p-group and γ 0 induces multiplications on both D ( T ) and T / D ( T ) . Recall that Σ is Aut ( A ) -isomorphic to r T . In the former case, that is if γ 0 PAut ( T ) , then γ 0 1 acts via conjugation on Σ as a power automorphism (that is a multiplication). In the latter case, Σ is critical as well and γ 0 1 induces invertible multiplications on both D ( Σ ) and Σ / D ( Σ ) . Thus, γ 0 1 acts via conjugation on Σ as an inertial automorphism of Σ, by Lemma 2.4 (5). In both cases, by Proposition 3.8, φ acts via conjugation on Σ as a finitary automorphism. Hence, γ 1 acts via conjugation on Σ as an inertial automorphism.

In the general case, when T is any periodic group and γ IAut ( T ) , then γ 1 (with respect to T K ) acts via conjugation as an inertial automorphism on all primary components Σ p of Σ, by what we have seen above and the fact that Σ p Hom ( A / T , A p ) . Similarly, since γ 1 acts as a multiplication on all but finitely many primary components A p of A, it acts the same way on all but finitely many Σ p . Thus, γ 1 is inertial on Σ by Lemma 2.4 (4).

It is clear that the action via conjugation of Γ 1 on Σ is faithful as the standard action of Γ 1 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 μ := α 1 (with respect to T K ) belongs to Γ 1 . If, by way of contradiction, Σ , μ is nilpotent-by-(locally finite), then there is s { 0 } such that Σ , μ s is nilpotent, so there is n such that [ Σ , n μ s ] = 0 , and hence 0 = Σ ( μ s - 1 ) n = Σ ( α s - 1 ) n . This is a contradiction, since Σ is unbounded as T is.

Finally, if the group A 2 is unbounded, then Σ 2 is unbounded as well. Let α be a non-periodic multiplication of A 2 . Then μ := α 1 1 with respect to the group A = A 2 A 2 K acts as non-periodic multiplication (by α) of Σ 2 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 IAut 1 ( A ) . Therefore, IAut 1 ( A ) / R is not locally finite. ∎

## Proof of Example B.

It follows from the last part of Theorem 6.2. ∎

Finally, we note that, despite the above propositions, in the general case the group IAut 1 ( A ) may be large.

Example 6.3

There exists an abelian group A with r 0 ( A ) = 1 and A p ( p ) for each prime p such that IAut ( A ) = IAut 1 ( A ) × { ± 1 } , IAut 1 ( A ) = Σ FAut ( A ) , where Σ := St IAut ( A ) ( A , T ( A ) ) FAut ( A ) with the property that Σ p ( p ) and IAut 1 ( A ) / FAut ( A ) Σ / T ( Σ ) is a divisible torsion-free abelian group with cardinality 2 0 .

Moreover, any element of IAut 1 ( A ) induces a finitary automorphism on T ( A ) .

## Proof.

As in [11, Proposition A], we consider the group G := B C , where B := p b p , C := p c p , and b p , c p have order p, p 2 , respectively, and p ranges over all primes. Consider the (aperiodic) element v := ( b p + p c p ) p G and V := v . We have that for each prime p there is an element d ( p ) G such that p d ( p ) = v - b p . Let A := V + d ( p ) | p and T := T ( A ) . Then

A / T 1 / p p ,

since A / T has torsion free rank 1 and v + T has p-height 1 for each p. Thus, T = T ( B ) p ( p ) and the p-component of A / V is generated by d ( p ) + V and has order p 2 , since p d ( p ) = v - b p .

Then Σ p ( p ) and Σ F Aut ( A ) = T ( Σ ) , hence Σ F Aut ( A ) . Moreover, A = d ( p ) + V , where V = v is infinite cyclic and A p = b p has order p. Also, Aut ( A / T ) = { ± 1 } and IAut ( A ) = IAut 1 ( A ) × { ± 1 } .

We claim that if γ IAut 1 ( A ) induces on T a finitary automorphism, then γ Σ FAut ( A ) . In fact, T γ is finite, so it is a π-component of A for some finite π. Thus, γ γ 0 - 1 Σ , where γ 0 := γ | A π 1 with respect to A = A π K and clearly γ 0 FAut ( A ) .

Finally, we prove the last part of the statement, from which it follows that IAut 1 ( A ) = Σ FAut ( A ) . To this end, let γ IAut 1 ( A ) and φ := γ - 1 . Since A φ T , there exists a non-zero integer n such that ( n v ) φ = 0 . We prove that T φ A π ( n ) , which is finite. For any prime p, on the one hand, n d ( p ) is a p-element modulo n v ker φ , hence ( n d ( p ) ) φ A p , that implies

( p n d ( p ) ) φ = p ( n d ( p ) ) φ = 0 .

On the other hand,

( p n d ( p ) ) φ = n ( v - b p ) φ = - n ( b p ) φ .

Hence, if p ( n ) φ , then A p φ = 0 . ∎

# Acknowledgements

We thank the referee for her/his useful comments.

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