The group of inertial automorphisms of an abelian group
aa r X i v : . [ m a t h . G R ] M a y The group of inertial automorphismsof an abelian group
Ulderico Dardano - Silvana Rinauro dedicated to Martin L. Newell
Abstract
We study the group IAut( A ) generated by the inertial automorphisms of an abeliangroup A , that is, automorphisms γ with the property that each subgroup H of A hasfinite index in the subgroup generated by H and Hγ . Clearly, IAut( A ) contains thegroup FAut( A ) of finitary automorphisms of A , which is known to be locally finite. Ina 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). In particular, IAut( A ) hasa normal subgroup Γ such that IAut( A ) / Γ is locally finite and Γ ′ is an abelian periodicsubgroup whose all subgroups are normal in Γ. In the case when A is periodic, IAut( A )results 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 thestructure of IAut( A ) in some other cases for A . Key words and phrases : finitary, commensurable, inert, locally finite, KI-groups. : Primary 20K30, Secondary 20E07, 20E36,20F24.
An endomorphism γ of an abelian group ( A, +) is called inertial endomorphism if andonly if ( H + Hγ ) /H is finite for each subgroup H (see [10], [12]). An inertial endomor-phism which is bijective is called inertial automorphism . This definition can be seen as ageneralization of the notion of finitary automorphism of A , that is, an automorphism γ acting as the identity map on a subgroup of finite index in A (see [2], [18]). Since A is belian, this condition is clearly equivalent to A ( γ −
1) being finite. Note that we regardabelian groups as right modules over their endomorphism ring and reserve the letter A for abelian groups, which are additively written .The concept of inertial endomorphism of an abelian group A may be used as a toolin the study of inert subgroups of possibly non-abelian groups (see [9], [11]). Recall thata subgroup is called inert if it is commensurable with its conjugates (see [1], [16]), wheresubgroups H and K are called commensurable if and only if H ∩ K has finite index inboth H and K .In this paper we study the group IAut( A ) generated by all inertial automorphismsof an abelian group A . Recall that in [7] (resp. [10]) we gave a description of inertialautomorphisms (resp. endomorphisms) of an abelian group, while the ring of inertialendomorphisms of A was featured in [8]. In particular, from [10], we have:- IAut( A ) consists of products γ γ − where γ and γ are both inertial automorphisms,- IAut( A ) is locally (central-by-finite),- IAut( A ) is abelian modulo its subgroup FAut( A ) of finitary automorphims.Recall that FAut( A ) is known to be locally finite ([18]).Note that the above definitions of IAut( G ) and FAut( G ) make sense even if the un-derlying group G is not abelian, and that FAut( G ) ≤ IAut( G ) in any case. Also, in [2]it has been shown that the group FAut( G ) of finitary automorphisms of any group G isboth abelian-by-(locally finite) and (locally finite)-by-abelian. Question . Is IAut( A ) , the group generated by all inertial automorphisms of an abeliangroup A , abelian-by-(locally finite)? Our main results, in Sect. 3, can be summarized as follows:- Theorem A gives a complete description of the group IAut( A ), when A is periodic;- Corollary A asserts that the answer to our question is in the positive, when A is periodic;- Theorem B asserts that, for any abelian group A , the group IAut( A ) has a metabeliansubgroup Γ such that IAut( A ) / Γ is locally finite and each subgroup of Γ ′ is normal in Γ;- Corollary B asserts that the answer is in the negative for A = Z ( p ∞ ) ⊕ Q p ;- Theorem C describes the group IAut( A ) in some cases when A is non-periodic.In our investigations, we shall look for abelian normal subgroups Σ of IAut( A ) suchthat the automorphisms induced by IAut( A ) via conjugation on Σ are inertial, see Theo-rem C. Thus, in Sect. 4, we highlight the role played by stability groups with respect tofinitary and inertial automorphisms of A . In Sect. 5 we treat the case when A is periodic(Theorem A). In Sect. 6 we treat the remaining cases (Theorems B and C). or undefined terminology, notation and basic facts we refer to [14] or [15]. In partic-ular, π ( n ) denotes the set of prime divisors of n ∈ Z . If π is a set of primes, then A π , T ( A ) and D ( A ) denote the unique maximum π -subgroup, the torsion subgroup and thedivisible subgroup of the abelian group A , respectively. By exponent m = exp ( A ) of a p -group A we mean the smallest m such that p m A = 0, or m = ∞ if A is unbounded.Furthermore, Q p is the additive group of rational numbers whose denominator is a powerof the prime p and Z ( p ∞ ) := Q p / Z . Also, r ( A ) denotes the torsion-free rank of A , i.e.the cardinality of a maximal Z -independent subset of A .If A i ≤ A and γ ∈ Aut( A ), as usual we denote by γ | A i the restriction of γ to A i . If wehave A = A ⊕ A and γ | A i = γ i ∈ Aut( A i ) (for i = 1 , γ = γ ⊕ γ .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 ] := aϕ − a = a ( ϕ − It is convenient to recall that an automorphism leaving every subgroup invariant is usuallycalled a power automorphism . Then the group PAut( A ) of power automorphisms of anabelian group A can be described as follows (see [15]).If A is a p -group and α = ∞ P i =0 α i p i (with 0 ≤ α i < p ) is an invertible p -adic, wedefine, with an abuse of notation, the power automorphism α (that we will also call multiplication by α ) by setting aα := ( k − P i =0 α i p i ) a , for any a ∈ A of order p k . In this way,we have defined an action on A of the group U p of units of the ring of p -adic integers, whoseimage is PAut( A ). If A has infinite exponent, then this action is faithful and PAut( A )is isomorphic to U p . Otherwise, if e := exp ( A ) < ∞ , then the kernel of this action is { α ∈ U p | α ≡ p e } and PAut( A ) is isomorphic to the group of units of Z ( p e ).If A is any periodic abelian group, then PAut( A ) is the cartesian product of all thePAut( A p ) where A p is the p -component of A . If A is non-periodic, then PAut( A ) = {± } .According to [10], an automorphism γ is called an (invertible) multiplication of A ifand only if it is a power automorphism of A , if A is periodic, or -when A is non-periodic-there exist coprime integers m, n such that ( na ) γ = ma , for each a ∈ A . In the lattercase, we have mnA = A and A π ( mn ) = 0 and -with an abuse of notation- we will write γ = m/n . We warn that we are using the word “multiplication” in a way differentfrom [14]. Invertible multiplications of A form a subgroup which is a central subgroup ofAut( A ). f r ( A ) < ∞ , from [10] we have that IAut( A ) contains the group of all invertible mul-tiplications. In this case, IAut( A ) consists of inertial automorphisms only. Furthermore,IAut( A ) is the kernel of the setwise action of Aut( A ) on the quotient of the lattice of thesubgroups of A with respect to commensurability (which is a lattice congruence indeed,since A is abelian).If r ( A ) = ∞ , then the above kernel is the subgroup of IAut( A ) consisting of so-called almost-power automorphisms of A , that is, automorphisms γ such that every subgroupcontains a γ -invariant subgroup of finite index. This group was introduced in [13] tostudy generalized soluble groups in which subnormal subgroups are normal-by-finite (orcore-finite, according to the terminology of [3] and [5]).We recall now some other facts that will be used in the sequel. They follow fromTheorem 3 of [7], Proposition 2.2 and Theorem A of [10]. Lemma 2.1.
Let γ be an automorphism of an abelian group A .1) If r ( A ) = ∞ , then γ is inertial if and only if there are a subgroup A of finite indexin A and an integer m such that γ = m on A .2) If < r ( A ) < ∞ , then γ is inertial if and only if there are a torsion-free γ -finitelygenerated γ -subgroup V such that A/V is periodic, a rational number m/n (with m , n coprime integers) such that γ = m/n on V and A π is bounded, where π := π ( mn ) . Inparticular A/A π is π -divisible.3) If A is periodic, then γ inertial if and only if γ is inertial on each p -component of A and acts as a power automorphism on all but finitely many of them.4) If A is a p -group, then γ is inertial if and only if either γ acts as an invertible mul-tiplication (that is as a power automorphism) on a subgroup A of finite index in A or(critical case) = D := D ( A ) has finite rank, A/D is infinite bounded and there is asubgroup A of finite index in A such that γ acts as invertible multiplication (by possiblydifferent p -adics) on both A /D and D . For further instances of inertial automorphisms see Lemma 6.1.
Our first result is about periodic abelian groups. heorem A. Let A be a periodic abelian group. Then there is a subgroup ∆ of IAut( A ) which is direct product of finite abelian groups and such that IAut( A ) = PAut( A ) · FAut( A ) · ∆ where ∆ is trivial, if A is reduced.Moreover, there are a set π of primes and subgroups Σ , Ψ of IAut( A ) such that Σ isan abelian π ′ -group with bounded primary components and FAut( A ) · ∆ = FAut( A π ) × (Σ ⋊ Ψ) where the automorphims induced by Ψ via conjugation on Σ are inertial and this actionis faithful. Corollary A. If A is a periodic abelian group, then IAut( A ) is central-by-(locally finite) .With an abuse of notation, in Theorem A we regard FAut( A π ) as naturally embedded inFAut( A ). For details in the case A is a p -group see Proposition 5.1 below.In the next theorem we answer our question in the non-periodic case. We reduce tostudy the subgroup IAut ( A ) consisting of inertial automorphisms of A that act as theidentity map on A/T ( A ). Actually, applying results from [7], we have that the aboveintroduced group of almost-power automorphisms of A isIAut ( A ) × {± } . In the case when r ( A ) = ∞ , we have IAut ( A ) = FAut( A ) by Lemma 2.1.(1).We will also consider a group Q ( A ) of particular inertial automorphisms of A , which iscointained in the center of Aut( A ) and is naturally isomorphic to the multiplicative groupof rational numbers generated by − p such that A/A p is p -divisible and A p is either bounded or finite according as r ( A ) is finite or not (see Lemma 6.1 below fordetails). Theorem B.
Let A be a non-periodic abelian group. Then there is a subgroup Q ( A ) ,which is isomorphic to a multiplicative group of rational numbers, such that IAut( A ) = IAut ( A ) × Q ( A ) Moreover there is a normal subgroup Γ of IAut ( A ) such that:i) IAut ( A ) / Γ is locally finite;ii) the derived subgroup Γ ′ of Γ is a periodic abelian group and each subgroup of Γ ′ isnormal in Γ orollary B. If A is an abelian group, then IAut( A ) is metabelian-by-(locally finite).However, IAut( Z ( p ∞ ) ⊕ Z ) is not nilpotent-by-(locally finite) .Note that if A is torsion-free, then IAut( A ) = Q ( A ) is abelian, as in Theorem 2 of [7].Further, in the statement of Theorem B one may take Γ to be the subgroup of IAut ( A )consisting of inertial automorphisms acting by multiplication on T ( A ). Unfortunatelythis subgroup need not be nilpotent, as in Corollary B. On the other hand, groups withproperty ( ii ) in Theorem B above have been studied under the name of KI-groups in aseries of papers (see [17]).The next theorem considers cases in which A splits on its torsion subgroup. For detailssee Propositions 6.3 and 6.4. Theorem C. If A is an abelian group with r ( A ) < ∞ and either T := T ( A ) is boundedor A/T is finitely generated, then there are subgroup Σ and Γ of IAut ( A ) such that IAut ( A ) = Σ ⋊ Γ where Σ is a periodic abelian group, Γ ≃ IAut ( T ) and the automorphisms induced by Γ via conjugation on Σ are inertial When
A/T is not finitely generated, it may happen that A has very few inertialautomorphisms, since from Proposition 6.2 and Lemma 2.1.(2) we haveIAut( Z ( p ∞ ) ⊕ Q ( p ) ) = {± } . However, in the general case, the group IAut ( A ) may be large, see Remark 6.5. We state now some basic facts that perhaps are already known (see also [4]). If X ≤ A ,we denote by St( A, X ) the stability group of the series A ≥ X ≥
0, that is, the set of γ ∈ Aut( A ) such that X ≥ [ A, γ ] := A ( γ −
1) and [
X, γ ] = 0. When X is a characteristicsubgroup of A , each γ ∈ Aut( A ) acts via conjugation on the abelian normal subgroupΣ := St( A, X ) of Aut( A ), according to the rule σ γ − σγ =: σ γ for each σ ∈ Σ.Similarly, γ acts on the additive group Hom( A/X, X ) of homomorphisms
A/X → X bya corresponding formula, i.e. ϕ γ − | A/X ϕγ where ϕ ∈ Hom(
A/X, X ) and γ | A/X denotesthe group isomorphism induced by γ on A/X . With an abuse of notation, we denote by σ − a ∈ A/X aσ − a ∈ X . act 4.1. The map H : σ ∈ St ( A, X ) ( σ − ∈ Hom(
A/X, X ) is an isomorphismof (right) Aut( A ) -modules, that is, for each γ ∈ Aut( A ) we have σ γ = γ − ( σ − γ + 1By this argument we have two technical lemmas. For the first one see [6] . Lemma 4.2.
Let A be an abelian group, σ, γ ∈ Aut( A ) and m , m ∈ Z . If σ stabilizes aseries ≤ A ≤ A , where γ = m on A and γ − = m on A/A , then σ γ = σ m m . Our next lemma deals with the case when A splits on X and will be used several times.In such a condition, once fixed a direct decomposition A = X ⊕ K , we have an embeddingAut( K ) → Aut( A ) given by γ ⊕ γ . Note that, if Γ ⊳ Aut( A ), then one can considerSt Γ ( A, X ) := St(
A, X ) ∩ Γ which is Aut( A )-isomorphic to a submodule of Hom( A/X, X ).The proof of the lemma is straightforward.
Lemma 4.3.
Let A = X ⊕ K , where X is a Γ -subgroup, Γ ≤ Aut( A ) , ζ : A/X ↔ K the natural isomorphism, Σ := St Γ ( A, X ) , Γ := { γ | X ⊕ | γ ∈ Γ } and Γ := { ⊕ ζ − γ | A/X ζ | γ ∈ Γ } . Then:1) if Γ ≤ Γ , then Γ = C Γ ( X ) ⋊ Γ and C Γ ( A/X ) = Σ ⋊ Γ ;2) if Γ ≤ Γ , then Γ = C Γ ( A/X ) ⋊ Γ and C Γ ( X ) = Σ ⋊ Γ ;3) if σ ∈ Σ , γ ∈ C Γ ( A/X ) and γ ∈ C Γ ( X ) , then σ γ γ = γ − ( σ − γ + 1 . In particular, if Γ Γ ≤ Γ , then Γ = Σ ⋊ (Γ × Γ ) . Proposition 4.4.
Let A be an abelian group and T := T ( A ) .1) If r ( A ) < ∞ , then the automorphisms induced by FAut( A ) via conjugation on St(
A, T ) are finitary;2) If r ( A ) = ∞ and the quotient A/T is free abelian, then there is γ ∈ FAut( A ) whichinduces 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 ). By Fact 4.1, for each σ ∈ St(
A, T ) wehave [ σ, γ ] H = ( σ − σ γ ) H = − ( σ − σ γ −
1) = − ( σ − σ − γ = ( σ − γ −
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 im ( ϕ σ ) ≤ im ( γ −
1) has finite order, say n . On the other hand, ker( ϕ σ ) ≥ n ¯ A and ¯ A/n ¯ A is finite since ¯ A has finite rank.2) If A = T ⊕ K , where K is free abelian on the infinite Z -basis { a i } , take γ ∈ FAut( T ) \ { } . Let t ∈ T such that tγ = t and γ := γ ⊕
1. For each i define σ i ∈ St(
A, T ) y 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 ] H ) ∋ a j for each i = j . (cid:3) Clearly, it may well happen that St(
A, T ) FAut( A ), as in the case A = Z ( p ∞ ) ⊕ Q p .On the other hand, we do have St( A, X ) ≤ FAut( A ), provided that one of the followingholds:- A/X is bounded and X has finite rank, as in Propositions 4.5 and 5.1.(2);- A/X has finite rank and X is bounded, as in Proposition 6.3;- A/X is finitely generated and X is periodic, as in Proposition 6.4.Here we consider an instance of the first case with X = D ( A ) and prove a propositionconcerning finitary automorphisms. Proposition 4.5.
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 actionis faithful. Proof.
First note that if σ ∈ Σ, then [
A, σ ] = A ( σ −
1) is finite, since it is both finiterank and bounded. Hence σ ∈ FAut( A ). Consider a decomposition A = D ⊕ B andapply Lemma 4.3, with X = D and Γ = FAut( A ) = C Γ ( X ). Put Φ := Γ . ThenFAut( A ) = Σ ⋊ Φ, as claimed.Let γ ∈ Φ. We have to show that set { [ σ, γ ] | σ ∈ Σ } is finite. As in Proposition4.4, we have [ σ, γ ] H = ( σ − σ γ ) H = (1 − σ ) + ( σ γ −
1) = − ( σ −
1) + γ − ( σ −
1) =( γ − − σ −
1) =: ϕ σ . Thus we have to count how many homomorphisms ϕ σ there are.On the one hand, ker( ϕ σ ) contains ker( γ − −
1) which has finite index in
A/D . On theother hand, the image of each ϕ σ is contained in the finite subgroup D [ p m ], where p m isa 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 maximalorder and b = bγ . Then B = h b i ⊕ B and we can write bγ = nb + b with n ∈ Z , b ∈ B .If b = nb , then there is σ ∈ Σ such that B ( σ −
1) = 0 and b ( σ −
1) = d where d ∈ D has the same order as b . Thus, by Fact 4.1, bγ ( σ γ −
1) = bγ ( γ − ( σ − d , while bγ ( σ −
1) = nd . Therefore σ γ = σ . Similarly, if b = nb , then there is σ ∈ Σ such that b ( σ −
1) = 0 and b ( σ −
1) = d of order p . Then bγ ( σ γ −
1) = 0, while bγ ( σ −
1) = d and again σ γ = σ . (cid:3) emark 4.6. In Proposition 4.5, Σ need not be contained in the FC-center of FAut( A ) . Proof.
Write A = D ⊕ B where D ≃ Z ( p ∞ ) and B = L i h b i i ≤ B is infinite andhomogeneous. Fix σ ∈ Σ such that b ( σ −
1) = d , where d is an element of D of order p ,and σ − D ⊕ L j =1 h b j i . For each i consider γ i ∈ FAut( A ) switching b i ↔ b andacting trivially on D ⊕ ( L j ,i } h b j i ). Then σ γi = γ − i ( σ −
1) + 1. Hence b i σ γ i = d + b i and b j σ γ i = b j for each j = i . (cid:3) Now an instance of a similar argument with X = T ( A ) Proposition 4.7.
Let A be an abelian group with r ( A ) < ∞ such that A/T is finitelygenerated (resp. T := T ( A ) is bounded). Then Σ := St(
A, T ) is a periodic (resp. bounded)abelian group and there is a subgroup Φ ≃ FAut( T ) such that FAut( A ) = Σ ⋊ Φ where Φ induces via conjugation on Σ finitary automorphims.If A/T = 0 is finitely generated, then this action is faithful, while if A = Z ⊕ Q (2) itis not. Proof.
In any case, we can write A = T ⊕ K where r := r ( K ) < ∞ . Recall thatΣ ≃ Hom(
A/T, T ). Note that Σ ≤ FAut( A ). In fact, if σ ∈ Σ, then σ − ∈ Hom(
A/T, T )and A ( σ −
1) is an abelian grougp 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 := { ϕ ⊕ | ϕ ∈ FAut( T ) } ≃ FAut( T ) and Φ ≤ FAut( A ). By Lemma4.3.(1) we have that FAut( A ) = Σ ⋊ Φ . By Proposition 4.4, Φ induces via conjugationon Σ finitary automorphisms.If A/T is finitely generated, then Σ ≃ Hom(
A/T, T ) is a periodic abelian group whichis naturally isomorphic to the direct sum of r copies of T as a right Aut( A )-module.Therefore the action of Φ on Σ is faithful. Finally, if A = Z ⊕ Q (2) , we have Φ ≃ U Z and Σ ≃ Z ; hence the action is not faithful. (cid:3) IAut( A ) , when A is periodic To give a detailed description of IAut( A ) when A a p -group, let us introduce some ter-minology. By 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 hat A = A ⊕ A ⊕ A where A is finite, exp ( A ) < e ≤ exp ( A ) and A is the sumof infinitely many cyclic groups of order p e . In [7] we called critical a p -group of type A = B ⊕ D with B infinite but bounded and D = 0 divisible with finite rank (see Lemma2.1.(4)). Critical groups will be a tool to describe IAut( A ) when A is periodic. Proposition 5.1.
Let A be an abelian p -group and D := D ( A ) .1) If A is non-critical, then IAut( A ) = PAut( A ) · FAut( A ) where PAut( A ) ∩ FAut( A ) iseither trivial or cyclic of order p m − e , according as A is unbounded or m := exp ( A ) < ∞ and e := eexp ( A ).2) If A = D ⊕ B is critical, let ∆ := { ⊕ n | n ∈ Z \ p Z } , Φ := { ⊕ ϕ | ϕ ∈ FAut( B ) } and Ψ := { ⊕ γ | γ ∈ IAut( B ) } , then IAut( A ) = PAut( A ) × (FAut A · ∆) . Moreover
FAut A · ∆ = C IAut ( A ) ( D ) = Σ ⋊ Ψ , where FAut( A ) = Σ ⋊ Φ andi) Σ := St(
A, D ) is an infinite abelian p -group, exp (Σ) = exp ( B ) =: m ′ < ∞ and eexp (Σ) = eexp ( B ) =: e ′ ;ii) Ψ = Φ∆ ≃ IAut( B ) where [Φ , ∆] = 1 and Ψ induces via conjugation on Σ inertialautomorphisms and this action is faithful;iii) ∆ ≃ PAut( B ) ≃ U ( Z ( p m ′ )) , each δ n := 1 ⊕ n ∈ ∆ acts via conjugation on Σ asthe multiplication by n and FAut( A ) ∩ ∆ has order p m ′ − e ′ ;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 2.1.(4), there exist a p -adic α anda subgroup A of finite index in A such that γ | A = α . Thus γ − α acts on A as theidentity map, that is, γ − α ∈ FAut( A ). Hence IAut( A ) = PAut( A ) · FAut( A ). Further,if the p -adic number β is in PAut( A ) ∩ FAut( A ), then β is trivial on a subgroup B offinite index in A . Therefore β = 1, provided exp ( A ) = ∞ . Otherwise, exp ( B ) ≥ e and β ≡ mod p e . Thus there are at most p m − e choices for such a β . On the other hand,each p -adic number β ≡ mod p e is finitary.2) Let A = D ⊕ B be critical. By Lemma 2.1.(4) there exists an invertible p -adic α such that γ | D = α . Thus γ := γα − ∈ C Γ ( D ). Clearly, PAut( A ) ∩ C Γ ( D ) = 1, so thatIAut( A ) = PAut( A ) × C Γ ( D ).Again by Lemma 2.1.(4), γ acts by multiplication by an integer n on a subgroup offinite index in A [ p m ′ ] where A [ p m ′ ] ≥ B . Therefore, if δ n := 1 ⊕ n ∈ ∆ with respect to A = D ⊕ B , we have γ δ − n ∈ FAut( A ). Hence C Γ ( D ) = FAut( A ) · ∆. t is routine to verify that ( i ) holds, since Σ := St( A, D ) ≃ Hom ( B, D ). By Proposi-tion 4.5, ( iv ) holds as well. By Lemma 4.3 (with X := D , K := B and so Γ = Ψ), wehave C Γ ( D ) = Σ ⋊ Ψ as stated in (2). Then, applying part (1) of the statement to B , wehave Ψ = ∆Φ and [Φ , ∆] = 1 as in ( ii ). Moreover, FAut( A ) ∩ ∆ has order p m ′ − e ′ .By Lemma 4.2, we have that ∆ acts on Σ as in ( iii ). Thus the whole Ψ = Φ∆ actsvia conjugation on Σ inducing inertial automorphisms and ( ii ) holds.It remains to show that Ψ acts faithfully on Σ. Let ϕδ n ∈ C Ψ (Σ) with ϕ ∈ Φ and δ n := 1 ⊕ n ∈ ∆. On the 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 δ n ∈ C Φ (Σ) = 1 by Proposition 4.5. (cid:3) We have seen that, if A is a p -group, then IAut( A ) is central-by-(locally finite). If A isa critical p -group, one can ask whether there is an abelian normal subgroup Λ of IAut( A )such that IAut( A ) = Λ · FAut( A ). The answer is in the negative, as in the followingremark. Fisrt we state an easy lemma. Lemma 5.2. If B is a subgroup of finite index in a bounded abelian group B , then thereare subgroups B and B such that B is finite, B ≥ B and B = B ⊕ B . Proof.
Clearly there is a finite subgroup F such that B = B + F . Since B is separableand B ∩ F is finite, then there is a finite subgroup B ≥ B ∩ F such that B = B ⊕ B for some B ≤ B . Fix B and B := B + F . On the one hand B + B = B + B + F = B + F = B . On the other hand, by Dedekind law, B ∩ B = B ∩ ( B + F ) = B ∩ ( B ∩ ( B + F )) = B ∩ ( B + ( B ∩ F )) = B ∩ B = 0. (cid:3) Remark 5.3. 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 5.1. Let n ∈ N be a primitiveroot of 1 mod p m ′ and consider δ := 1 ⊕ n ∈ ∆ with respect to A = D ⊕ B . Since∆ ≤ C Γ ( D ) = FAut( A ) · Λ, then we can write with ϕ ∈ FAut( A ) and λ ∈ Λ. Hence δ = λ = n on some subgroup B of finite index in B . By Lemma 5.2, B = B ⊕ B with B ≤ B and B finite. Put A := D + B and note that λ | A = 1 ⊕ n with respect to A = D ⊕ B .It is sufficient to show that h λ i Γ is infinite and not locally nilpotent, where Γ is thegroup of (inertial) automorphisms of A of type γ ⊕ A = A ⊕ B , with ∈ IAut( A ). Thus we may assume A = A and Γ := Γ . Then multiplication by n isin Λ and Λ = ∆ Γ .We claim that ∆ Γ = Σ ⋊ ∆. In fact, by Proposition 5.1 we have that ∆ ≃ U ( Z p m ′ )acts faithfully by multiplication on the infinite abelian p -group Σ of exponent m ′ and thenΣ = [Σ , ∆] and ∆ Γ = Σ∆, as claimed. Thus ∆ Γ is not locally nilpotent, since the actionof ∆ on Σ is fixed-point-free. (cid:3) Proof of Theorem A . By Lemma 2.1.(3), IAut( A ) may be identified with PAut( A ) · Dr p IAut( A p ). Apply Proposition 5.1 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, foreach p π , there are subgroups ∆ p , Σ p , Ψ p corresponding to ∆ , Σ , Ψ in Proposition 5.1such that IAut( A p ) = PAut( A p ) · FAut( A p ) · ∆ p and FAut( A p ) · ∆ p = Σ p ⋊ Ψ p . Now it isroutine to verify that the statement follows by setting ∆ := Dr p π ∆ p , Σ := Dr p π Σ p ,Ψ := Dr p π Ψ p , and recalling that Dr p FAut( A p ) = FAut( Dr p A p ).Remark that, in Theorem A, when we consider the action of the above Ψ on the p -component Σ p of 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. IAut( A ) , when A is non-periodic To prove Theorem B we point out the existence of some inertial automorphisms of aparticular type.
Lemma 6.1.
Let A be a non-periodic abelian group and π ∗ ( A ) be the set of primes suchthat A/A p is p -divisible and one of the following holds:- A p is finite,- r ( A ) is finite and A p is bounded.Then, for each p ∈ π ∗ ( A ) , there is a unique C ( p ) such that A = A p ⊕ C ( p ) and theautomorphism γ ( p ) := 1 ⊕ p (with respect to this decomposition) is inertial.Moreover, the subgroup Q ( A ) := h γ ( p ) | p ∈ π ∗ ( A ) i × {± } is a central subgroup of IAut( A ) which is isomorphic to a multiplicative group of rational numbers. Furthermore, IAut ( A ) ∩ Q ( A ) = 1 . Proof.
The proof of the first part of the statement, concerning C ( p ) , is routine. Letus show that γ ( p ) is inertial. For each H ≤ A we have H + Hγ ( p ) ≤ H + A p . If A p isfinite, then ( H + Hγ ( p ) ) /H is finite as well and γ ( p ) is inertial. If r ( A ) is finite and A p s bounded, let V be the h γ ( p ) i -closure of a free subgroup of C with maximal rank. Then V is torsion-free, C/V is a p ′ -group and γ ( p ) acts as a power automorphism on A/V .Thus we apply Lemma 2.1.(2) and deduce that γ ( p ) is inertial. The remaining part of thestatement follows straightforward. (cid:3) Proof of Theorem B . Let γ ( p ) and Q := Q ( A ) as in Lemma 6.1.We first consider the case when r ( A ) = ∞ . Let γ ∈ IAut( A ). Then, by CorollaryB in [10], we have γ = γ γ − with γ , γ inertial. Further, by Lemma 2.1.(1), there isa subgroup A with finite index in A such that we have γ | A = m/n = p s · · · p s t t ∈ Q ( m, n coprime, p i prime, s i ∈ Z ). Also IAut ( A ) = FAut( A ) and γ = m/n on A/T aswell. If m/n = 1, then γ ∈ FAut( A ). If m/n = −
1, put γ := − ∈ Q . Otherwise, since γ is invertible, mA = A = nA . Then for each p i ∈ π := π ( mn ), the p i -componentof A is finite and A/T is p i -divisible. Consider then γ := γ s ( p ) · · · γ s t ( p t ) ∈ Q . In bothcases, γγ − = 1 on A / ( A ) π hence γγ − ∈ FAut( A ). Thus IAut( A ) = IAut ( A ) × Q ( A ).Moreover, ( i ) and ( ii ) are true with Γ = 1, since IAut ( A ) = FAut( A ) is locally finite.Let then r ( A ) < ∞ and γ ∈ IAut( A ). By Corollary B in [10] γ is inertial. By Lemma2.1.(2), we have that γ = m/n = p s · · · p s t t ∈ Q ( m, n coprime, p i prime, s i ∈ Z ) on A/T .We also have that, for each p i ∈ π := π ( mn ), the group A/T is p i -divisible and A p i isbounded. Consider γ := γ s ( p ) · · · γ s t ( p t ) ∈ Q , Clearly γ = m/n on A/T . Thus γγ − actstrivially on A/T and IAut( A ) = IAut ( A ) × Q ( A ), as stated.Let Γ be the preimage of PAut( T ) under the canonical homomorphism IAut ( A ) IAut( T ). Now ( i ) holds, since IAut ( A ) / Γ is locally finite by Theorem A. To check( ii ) consider that the derived subgroup Γ ′ of Γ stabilizes the series 0 ≤ T ≤ A andtherefore is abelian. Moreover, by Theorem B in [10], the subgroup Γ ′ consists of finitaryautomorphisms. Thus Γ ′ is torsion and ( ii ) holds by Lemma 4.2.Let us see that there are groups A with few inertial automorphisms even if r ( A ) < ∞ and that the canonical homomorphism IAut ( A ) → IAut( T ) need not be surjective. Proposition 6.2.
Let A be a π -divisible non-periodic abelian group, where π is a set ofprimes. If T := T ( A ) is a π -group, then IAut ( A ) = 1 . Proof. If r ( A ) = ∞ then IAut ( A ) = FAut( A ). Moreover, if γ ∈ FAut( A ) then A ( γ − π -group. Then A/ ker( γ −
1) is such. Hence A = ker( γ −
1) and FAut( A ) = 1.If r ( A ) < ∞ , by Lemma 2.1.(2) 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 2.1, part (3) and (4), we have that γ is a multiplication on B/V . Furthermore, he 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 − A/B → T which is necessarily 0 since A/B isa π ′ -group. Thus γ = 1 on the whole group A . (cid:3) Proof of Theorem C . It follows from the next propositions which considers cases inwhich IAut ( A ) splits on Σ := St( A, T ). (cid:3) Proposition 6.3.
Let A be an abelian group and T := T ( A ) .If r ( A ) < ∞ and T is bounded, then Σ := St(
A, T ) is a bounded abelian group andthere is a subgroup Γ of IAut ( A ) such that Γ ≃ IAut( T ) and IAut ( A ) = Σ ⋊ Γ where Γ induces via conjugation on Σ inertial automorphisms. Proof.
We can write A = T ⊕ K where r := r ( K ) < ∞ . Note that the groupΣ ≃ Hom(
A/T, T ) is a periodic abelian group which is bounded as T .Clearly Γ := { γ ⊕ | γ ∈ IAut( T ) } ≃ IAut( T ). If γ ∈ IAut( T ), then γ ⊕ T ⊕ K ) is inertial by Lemma 2.1.(4), and so Γ ≤ IAut ( A ). Thus we mayapply Lemma 4.3 with Γ := IAut ( A ). We obtain IAut ( A ) = Σ ⋊ Γ , as claimed.By Proposition 5.1, we have IAut( T ) = FAut( T ) · PAut( T ). Hence Γ = Φ ∆ whereΦ := { ϕ ⊕ | ϕ ∈ FAut( T ) } ≃ FAut( T ) acts conjugation on Σ by means of finitaryautomorphisms, by Proposition 4.7 and ∆ := { δ ⊕ | δ ∈ PAut( T ) } ≃ PAut( T ) acts viaconjugation on Σ by means of multiplications, by Lemma 4.2. Therefore the whole Γ induces via conjugation on Σ inertial automorphisms. (cid:3) We notice that the action of Γ on Σ in Proposition 6.3 need not be faithful, as alreadyseen in Proposition 4.7. Proposition 6.4.
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 thereis a subgroup Γ of IAut ( A ) such that Γ ≃ IAut( T ) and IAut ( A ) = Σ ⋊ Γ where Γ induces via conjugation on Σ inertial automorphisms and this action is faithful.If in addition T is unbounded, then IAut ( A ) is not nilpotent-by-(locally finite). Fur-ther, if A ′ is unbounded, then IAut ( A ) is not even (locally nilpotent)-by-(locally finite). roof. As in the proof of Propositon 6.3, we can write A = T ⊕ K where K is finitelygenerated. The group Σ ≃ Hom(
A/T, T ) is a periodic abelian group which is isomorphicto the direct sum ⊕ r T of r := r ( A ) > T as a right Aut( A )-module.Clearly Γ := { γ ⊕ | γ ∈ IAut( T ) } ≃ IAut( T ). If γ ∈ IAut( T ), then γ ⊕ T ⊕ K ) is inertial by Lemma 2.1.(4). Hence Γ ≤ IAut ( A ). Thus we mayapply Lemma 4.3 with Γ := IAut ( A ), and we obtain IAut ( A ) = Σ ⋊ Γ . Let us investigate now the action of Γ via conjugation on Σ. Assume first that T is a p -group. Let γ ∈ IAut( T ). By Proposition 5.1, γ = γ ϕ , where ϕ ∈ FAut( T ) andeither γ ∈ PAut( T ) or T is a critical p -group and γ induces multiplications on both D ( T ) and T /D ( T ). Recall that Σ is Aut( A )-isomorphic to ⊕ r T . In the former case, thatis if γ ∈ PAut( T ), then γ ⊕ γ ⊕ D (Σ) and Σ /D (Σ). Thus γ ⊕ ϕ actsvia conjugation on Σ as a finitary automorphism. Hence γ ⊕ T is any periodic group and γ ∈ IAut( T ), then γ ⊕ T ⊕ K ) acts via conjugation as an inertial automorphism on all primarycomponents Σ p of Σ, by what we have seen above and the fact that Σ p ≃ Hom(
A/T, A p ).Similarly, since γ ⊕ A p of A , it acts the same way on all but finitely many Σ p . Thus γ ⊕ on Σ is faithful as the standard actionof Γ on T is such.To prove the last part of the statement, note that in the case when T is unbounded,then there exists a non-periodic multiplication α of T . Note that the automorphism µ := α ⊕ T ⊕ K ) belongs to Γ . If, by the way of contradiction, h Σ , µ i is nilpotent-by-(locally finite), then there is s ∈ Z \ { } such that h Σ , µ s i is nilpotent,so there is n ∈ N such that [Σ , n µ s ] = 0, and hence 0 = Σ ( µ s − n = Σ ( α s − n . This is acontradiction, since Σ is unbounded as T is.Finally, if A ′ is unbounded, then Σ ′ is unbounded as well. Let α be a non-periodicmultiplication of A ′ . Then, µ := α ⊕ ⊕ A = A ′ ⊕ A ⊕ K acts asnon-periodic multiplication (by α ) of Σ ′ 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 nilpotentradical R of IAut ( A ). Therefore IAut ( A ) /R is not locally finite. (cid:3) inally, we note that, despite the above propositions, in the general case the groupIAut ( A ) may be large. Remark 6.5.
There exists an abelian group A with r ( A ) = 1 and A p ≃ Z ( p ) for eachprime p such that IAut( A ) = IAut ( A ) × {± } , IAut ( A ) = Σ · FAut( A ) , where Σ :=StIAut ( A ) ( A, T ( A )) FAut( A ) , Σ ≃ Q p Z ( p ) and IAut ( A ) / FAut( A ) ≃ Σ /T (Σ) is adivisible torsion-free abelian group with cardinality ℵ .Moreover any element of IAut ( A ) induces on T a finitary automorphism. Proof.
As in Proposition A in [10], we consider the group G := B ⊕ C where B := Q p h b p i , C := Q p h c p i , and b p , c p have order p , p resp. and p ranges over all primes. Considerthen the (aperiodic) element v := ( b p + pc p ) p ∈ G and V := h v i . We have that for eachprime p there is an element d ( p ) ∈ G such that pd ( p ) = v − b p . Let A := V + h d ( p ) | p i .Then A/T ≃ h /p | p i ≤ Q , since A/T has torsion free rank 1 and v + T has p -height1 for each p . Thus T = T ( B ) ≃ L p Z ( p ) and the p -component of A/V is generated by d ( p ) + V and has order p , since pd ( p ) = v − b p .Then Σ ≃ Q p Z ( p ) and Σ ∩ F Aut( A ) = T (Σ), hence Σ F Aut( A ). Moreover A = h d ( p ) i + V , where V = h v i is infinite cyclic and A p = h b p i has order p . AlsoAut( A/T ) = {± } and IAut( A ) = IAut ( A ) × {± } .We claim that if γ ∈ IAut ( 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 γγ − ∈ Σ, where γ := γ | A π ⊕ A = A π ⊕ K and clearly γ ∈ FAut( A ).Finally we prove the last part of the statement, from which it follows IAut ( A ) =Σ · FAut( A ). Let γ ∈ IAut ( A ) and ϕ := γ −
1. Since Aϕ ≤ T , there exists an integer n = 0 such that ( nv ) ϕ = 0. We prove that T ϕ ⊆ A π ( n ) , which is finite. For any prime p , on the one hand, nd ( p ) is a p -element modulo h nv i ≤ ker ϕ , hence ( nd ( p ) ) ϕ ∈ A p , thatimplies ( pnd ( p ) ) ϕ = p ( nd ( p ) ) ϕ = 0. On the other hand, ( pnd ( p ) ) ϕ = n ( v − b p ) ϕ = − n ( b p ) ϕ .Hence, if p ( n ) ϕ , then A p ϕ = 0. (cid:3) References [1] V. V. Belyaev, M. Kuzucuoglu, E. Seckin, Totally inert groups.
Rend. Sem. Mat.Univ. Padova (1999), 151-156.[2] V. V. Belyaev, D. A. Shved, Finitary automorphisms of groups.
Proc. Steklov Inst.Math. , suppl. 1 (2009), S49-S56.
3] J. T. Buckley, J.C. Lennox, B. H. Neumann, H. Smith, J. Wiegold, Groups with allsubgroups normal-by-finite.
J. Austral. Math. Soc.
Ser. A , no. 3 (1995), 384-398.[4] C. Casolo, O. Puglisi, Hirsch-Plotkin radical of stability groups. J. Algebra (2012), 133-151.[5] G. Cutolo, E.I. Khukhro, J.C. Lennox, S. Rinauro, H. Smith, H, J. Wiegold, Locallyfinite groups all of whose subgroups are boundedly finite over their cores.
Bull. LondonMath. Soc. , no. 5 (1997), 563-570.[6] U. Dardano, C. Franchi, A note on groups paralyzing a subgroup series. Rend. Circ.Mat. Palermo (2) , no. 1 (2001), 165-170.[7] U. Dardano, S. Rinauro, Inertial automorphisms of an abelian group, Rend. Sem.Mat. Univ. Padova (2012), 213-233.[8] U. Dardano, S. Rinauro, On the ring of inertial endomorphisms of an abelian group.
Ricerche Mat. , no. 1 suppl. (2014), S103-S115.[9] U. Dardano, S. Rinauro, On groups whose subnormal subgroups are inert, Int. J.Group Theory , , no. 2 (2015), 17-24.[10] U. Dardano, S. Rinauro, Inertial endomorphisms of an abelian group, Ann. Mat.Pura Appl. , to appear, DOI: 10.1007/s10231-014-0459-6, also see arXiv:1310.4625[11] D. Dikranjan, A. Giordano Bruno, L. Salce, S. Virili, Fully inert subgroups of divisibleAbelian groups,
J. Group Theory (2013), 915-939.[12] D. Dikranjan, A. Giordano Bruno, L. Salce, S. Virili, Intrinsic algebraic entropy, J.Pure Appl. Algebra (2015), 2933-2961.[13] S. Franciosi, F. de Giovanni and M.L. Newell, Groups whose subnormal subgroupsare normal-by-finite,
Comm. Alg. , no. 14 (1995), 5483-5497.[14] L. Fuchs, “Infinite Abelian Groups”, Academic Press, New York - London, 1970-1973.[15] D.J.S. Robinson, “A Course in the Theory of Groups”, 2nd. ed., Grad. Texts inMath. , Springer V., New York 1996[16] D.J.S. Robinson, On inert subgroups of a group, Rend. Sem. Mat. Univ. Padova (2006), 137-159.[17] I. Ya. Subbotin, On the ZD-coradical of a KI-group. (Russian)
Vychisl. Prikl. Mat.(Kiev) (1991), 120–124; translation in J. Math. Sci. , no. 3 (1994), 3149-3151.[18] B.A.F. Wehrfritz, Finite-finitary groups of automorphisms. J. Algebra Appl. , no. 4(2002), 375-389. lderico Dardano, Dipartimento di Matematica e Applicazioni “R.Caccioppoli”, Uni-versit`a di Napoli “Federico II”, Via Cintia - Monte S. Angelo, I-80126 Napoli, Italy. [email protected] Silvana Rinauro, Dipartimento di Matematica, Informatica ed Economia, Universit`a dellaBasilicata, Via dell’Ateneo Lucano 10, I-85100 Potenza, Italy. [email protected]@unibas.it