Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

Abstract Necessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.


Introduction
All groups mentioned in this paper are Abelian groups and are written additively, unless otherwise specified.
The concept of entropy was first introduced to Abelian groups in 1965 in the paper [1]. In that paper, the main idea was sketched in the final section. Later on, Weiss [21] continued the study of entropy in groups and established many basic properties of such an entropy. In 1979 Peters [16] defined a different kind of entropy for automorphisms of discrete groups by taking finite subsets instead of finite subgroups; for torsion groups the resulting entropy is the same as that outlined in [1].
In recent years various "algebraic" entropies in groups have been studied intensively. D. Dikranjan et al in [7] developed algebraic entropy based on Weiss's definition. In that paper many basic results, including the important Addition Theorem and the Poincaré-Birkhoff recurrence theorem for algebraic entropy, were established. The note following the recurrence theorem [7, Proposition 2.9] is of interest; it states that a group of zero algebraic entropy is necessarily co-Hopfian. This is an important connection between entropy and (co-)Hopficity. Later on, D. Dikranjan, A. Giordano and L. Salce [6], and B. Goldsmith and K. Gong [9] introduced another kind of entropy in groups; they call such entropy, the adjoint (algebraic) entropy. In these papers finite index subgroups were used instead of finite subgroups, and pre-images instead of images. It is interesting to recall an observation in [9,Corollary 2.22]: a reduced torsion-free group with zero adjoint entropy is necessarily Hopfian; this corollary connects the notions of adjoint entropy and Hopficity in a similar fashion to the connection between algebraic entropy and co-Hopficity.
Various developments of "algebraic" entropies were made meanwhile; for instance, in [19], groups with zero adjoint algebraic entropy were studied in full, while entropies for modules over various classes of rings, have been considered in [22], [18].
As mentioned above, algebraic entropy and adjoint algebraic entropy have close connections with the concepts of co-Hopficity and Hopficity. Recall the definitions of Hopficity and co-Hopficity: a group is called Hopfian if every surjective endomorphism is an automorphism, and co-Hopfian if every injective endomorphism is an automorphism; or equivalently, a group is called Hopfian it has no proper isomorphic quotient groups and co-Hopfian if it has no proper isomorphic subgroups. These notions were introduced by Baer [2] who used the terminology Q-groups and S-groups, where Q-groups were exactly the Hopfian groups and S-groups were co-Hopfian groups. It should be noted that such properties are defined for arbitrary (noncommutative) groups and it was in that context that Baer initiated their study.
An immediate question which arises is the easily posed, but not so easily answered, "Is the direct sum of two (co-)Hopfian groups again (co-)Hopfian?". This question for Hopfian groups was mentioned in [3] and was immediately answered for Abelian groups by A. L. S. Corner [4], who constructed two torsion-free Abelian Hopfian groups which have non-Hopfian direct sum. Recently Vámos and the first author [11] have shown that for any positive integer n, there exists a torsion-free Hopfian group G such that the direct sum of n copies of G is Hopfian but the direct sum of n + 1 copies is not.
In the context of arbitrary groups Hirshon [14] has established many interesting results which guarantee the Hopficity of a direct product of groups: for example, if every proper homomorphic image of A is Abelian and B is a group which satisfies the ascending chain condition for normal subgroups, then A × B is Hopfian. For arbitrary co-Hopfian groups, Li [15] has established that the direct product of two such groups A, B is again co-Hopfian if the pair is semi-rigid in the sense that either there are no non-trivial homomorphisms from A to B or from B to A. In Section 4 we shall give simple matrix-based proofs for some analogous results in the Abelian context.
Due to the close connections between (co-)Hopficity and the algebraic entropy and adjoint algebraic entropy, it is natural to ask if similar "closure" properties hold for direct sums of Abelian groups with zero (adjoint) algebraic entropy. We shall address such questions in Section 3.
We finish this introduction by noting that our notation is standard and follows that in Fuchs [8].

Definitions and Basic results
In this section, we recall the definitions of algebraic entropy and adjoint entropy of groups, and list some useful properties of such entropies. Some basic propositions and auxiliary lemmas are then stated for later use in Section 3. Their proofs can be derived easily from the references given, so proofs will be omitted or only an outline sketch will be provided. Recall the definition of algebraic entropy, see [7]: let G be a group, ϕ an endomorphism of G, then for every positive integer n and every finite subgroup F of G, set The Tn(ϕ, F) is called the n th -trajectory of F respect to ϕ, and the subgroup T(ϕ, F) = ∑︀ n≥0 ϕ n F is called the trajectory of F respect to ϕ. Denote the cardinality of Tn(ϕ, F) by |Tn(ϕ, F)|; it was shown in [7] that the following limit does exist We mention the following proposition, which appears as [ This fact together with the so-called Addition Theorem, see [7,Theorem 3.1] and the fact that the socle of a group is fully invariant give the first part of the following useful corollary: We now consider adjoint entropy. Recall the definition, see, for example, [6,9]: let G be a group and N a finite index subgroup of G, ϕ an endomorphism of G, for every fixed natural number n set It is pointed out in papers [6,9] that Cn(ϕ, N) is a finite index subgroup in G. Cn(ϕ, N) is called the n th -cotrajectory of N with respect to ϕ, and the subgroup C(ϕ, N) = ∩ n≥0 ϕ −n N is called the co-trajectory of N with respect to ϕ. Denote log |G/Cn(ϕ, N)| by In(ϕ, N), then the following limit exists as shown in, for instance, [6,9]: This entropy has many nice properties; here we mention a few which will be useful later. These properties are contained in [6,Corollary 7.7], but we prefer to re-state part of that corollary here for later use: From this result, the following corollary is immediate:

Corollary 2.4. For a group A, if A/pA is finite for every prime number p, then ent * (A) = 0. In particular, when A is torsion and each primary component is finite, then ent * (A) = 0; and if A is torsion-free of finite rank, then
The following result is well known and will be used repeatedly in the sequel. This theorem guarantees that there is no confusion when we say a matrix of this kind is an endomorphism of a direct sum of two groups or vice versa. The next fact is straightforward:

Lemma 2.6. For groups A and B, and homomorphisms ϕ and ψ from A to B, if C is a subgroup of B, then
The final result in this section concerns a fundamental property of the so-called small homomorphisms of p-groups; recall the definition: if A, C are p-groups, then a homomorphism ϕ : A → C is said to be small if, given any positive integer e, there exists a positive integer m such that ϕ((p m A)[p e ]) = 0. The basic result that we shall need later is the well-known: Bn, where each Bn is a direct sum of cyclic groups of order p n , is a basic subgroup of A and ϕ is a small homomorphism ϕ : and y ∈ Hm [p]. A straightforward calculation shows that a socle element of Hm must have height (in A) at least p m and hence ϕ(y) = 0. So

Direct sums of groups with zero entropy
The principal objective in this section is to find reasonably general sufficient conditions that will ensure that the direct sum of two groups with zero entropy is again a group of zero entropy. We begin with the situation for p-groups of zero algebraic entropy.
Recall that a homomorphism ϕ from a group A to a group B is called socle-finite if the image of the socle of A is finite.
. Note that A is invariant under ϕ and so by the Addition Theorem [7, Theorem 3.1], ent(ϕ) = ent(ϕ A) + ent(φ), where the latter is the induced mapping on the quotient A ⊕ B/A. Since ent(A) = 0, the first term in the equation above is zero; the second term is also zero sinceφ is conjugate to an endomorphism of B and ent(B) = 0.
Consider an arbitrary but fixed finite subgroup F of (A ⊕ B) [p]. If we show that the trajectory of F with respect to Φ is finite, then Φ (A ⊕ B)[p] has zero algebraic entropy and it will follow from Corollary 2.2 that Φ has zero algebraic entropy.
Since F is a finite subgroup of ( . . Now since we have already proved that ent(ϕ) = 0, thus, by Corollary is also finite. Hence the proof is complete. Remark. We recall a well-known realization theorem of Corner [5, Theorem 1.1], which states that there is a family of 2 2 ℵ 0 groups G i such that every homomorphism between different groups of the family is small; furthermore, it is proved in [7,Thoerem 5.4] that there is a family of 2 2 ℵ 0 groups G i such that every group of the family has zero algebraic entropy and that each homomorphism between distinct members of the family is small. With these two facts and Corollary 3.2, one can deduce that there is a family of 2 2 ℵ 0 groups G i , each having zero algebraic entropy and such that the direct sum G i ⊕ G j (i ≠ j) again has zero algebraic entropy.

Adjoint entropy
The arguments in this section are in some sense dual to those for algebraic entropy but involve concepts that are not so well known in the literature. First we make a rather ad hoc definition: given a prime p, groups A, B and a homomorphism ϕ : B → A, we say that ϕ is p-quotient finite if the induced mapφp : B/pB → A/pA has finite image. The mapping ϕ is said to be quotient-finite if it is p-quotient finite for all primes p. It is straightforward to show that ϕ is quotient-finite if ϕ −1 (pA) is of finite index in B for all primes p.
The main result in this subsection is:  N A ⊕ N B ) is also finite index in A ⊕ B and stationary eventually, and since N is arbitrarily chosen, then from [9, Lemma 2.7] and its proof, or from [6, Proposition 2.3] one can deduce that the adjoint entropy of Ψ is zero, and therefore the proof is complete. We first show that ent * (ϕp) = 0. Note that by the assumption and analysis, ent * (α) = ent * (β) = 0.
Notice that θ −1 L ≥ Kerθ, and the latter is a finite index subgroup of A ⊕ B: in fact, a standard verification shows that Kerθ ≥ A ⊕ Ker . By the condition that is quotient-finite we deduce that the kernel of is a finite index subgroup of B, thus A⊕Ker is of finite index in A⊕B and so is Kerθ. Notice that ϕ −1 L ≥ C(ϕ, L) and therefore, Ψ −1 L ≥ C(ϕ, L) ∩ Kerθ. Furthermore, Ψ −2 L ≥ (ϕ + θ) −1 (C(ϕ, L) ∩ Kerθ), again by Lemma 2.6, and so we have

3], this gives that C(ϕ, M) is also a finite index subgroup of
We continue in the same way and deduce: M). In other words, the cotrajectory of Ψ with respect to L = N A + N B is stationary eventually. This completes the proof.

Corollary 3.4. If A is a group of zero adjoint entropy and B is group such that B/pB is finite for all primes p, then A ⊕ B again has zero adjoint entropy. In particular, if B is finite or B is torsion-free of finite rank, then A ⊕ B has zero adjoint entropy.
Proof. If B/pB is finite for all primes p, then it follows from Corollary 2.4 that ent * (B) = 0. The finiteness of B/pB for all primes p ensures that every map from B → A is quotient-finite. The particular cases follow easily since in each case it is clear that B/pB is finite for all primes p.
We remark that the condition that B/pB be finite does not restrict one to groups of finite rank; for example, it is a well-known result of Griffith [13] that a torsion-free group B has the property that rp(B) = dim Z/pZ (B/pB) ≤ 1 if, and only if, B is isomorphic to a pure subgroup of the group J = ∏︀ p Jp.

Direct sums of (co-)Hopfian groups
In this section we investigate the (co-)Hopficity of a direct sum sum of certain (co-)Hopfian groups. The results are similar to those obtained in the previous section for groups of zero entropy, but we note that the results we obtain here are not immediate corollaries of those in the previous section. Some results in this section were previously obtained by Hirshon [14] and Li [15]. Here in this note proofs to these results are given in a systematic way by using representative of a homomorphism of matrix. This approach is transparent and natural.
We recall for later purposes that there is a well-known strong hypothesis which ensures (co-)Hopficity of a direct sum as the following proposition. It can be considered as a special case of [ We begin by investigating sums of Hopfian groups; as heretofore the groups are additively written Abelian groups.
We first investigate the Hopficity of a direct sum of a Hopfian group and a cyclic p-group. The first observation is Proof. Without loss in generality assume α is surjective. Since A is a Hopfian group and α is epic on A, then α is invertible. Note that for any δ, To prove ∆ is invertible, it suffices to show that β − α −1 δ is surjective since the Hopficity of B would imply it is invertible. To see this, pick any element b ∈ B. By the surjectivity of ∆ we have elements Thus we see the following identities we see that ∆ is invertible and so is ϕ.
For the later reference we list: Proof. Without loss in generality it suffices to handle the case when δ = 0. We claim that α is surjective and thus by Proposition 4.2 ϕ is invertible. For any element a ∈ A, since ϕ is surjective, then there exists elements x ∈ A, y ∈ B such that ϕ From the first equation and the arbitrariness of a, we see α is surjective.
We can deduce the following: (see [14]). Suppose that A is a Hopfian group and B is a cyclic p-group generated by the element b, then the direct sum G = A ⊕ B is also Hopfian.
where β is an integer. We divide the proof into two cases: i) At least one of α, β1 B is surjective. In this case, by Proposition 4.2, ϕ is invertible. ii) Both α, β1 B are not surjective. It is easy to see that p divides β. Since ϕ is surjective, then we have the necessary condition: Since α is not surjective, we claim that δ(b) ∉ α(A). For if δ(b) ∈ α(A), then, since δ(B) is generated by  δ(b), δ(B) ≤ α(A) forcing α(A) = A, contrary to the fact that α is not surjective. Thus we can write α(A)∩ δ(B) = ⟨︀ p s δ(b) ⟩︀ , s ≥ 1. Since is surjective. Thus, we have the following necessary condition: Since ϕ is surjective, there exist elements x ∈ A, λb ∈ B such that ϕ On the other hand, p divides 1 − λ and p divides β, thus (1 − λ) − βλ + 1 is relatively prime to p, and hence is an automorphism of the cyclic p-group α(A) ∩ δ(B). Hence δ(b) ∈ (α + δ )(A) and by the necessary condition (1), α + δ is a surjection. Thus it is invertible. Hence ∆ is invertible by Proposition 4.2, and so is ϕ By induction, we deduce the following Theorem 4.5. (see [14]). Suppose that A is a Hopfian group and B is a finite group, then the direct sum G = A⊕B is also Hopfian.
We can extend Theorem 4.5, assuming the group B finitely generated; it is worthwhile to note that finitely generated groups are Hopfian (this is true for finitely generated modules over any commutative ring, by a celebrated result by Vasconcelos, see [20]. Theorem 4.6. (see [14]). If A is Hopfian, and B is finitely generated, then A ⊕ B is Hopfian.
Now it is clear that µ : A−→A with µ(x + ka) = α(x) + ka ′ (k any integer) is an endomorphism of A and in fact it is a surjection. By hopficity of A we see µ is an automorphism; note that µ Ker = α Ker . Thus Ker ∩ Kerα = {0}, so in case i), Kerϕ = 0 and hence ϕ is an automorphism.

Co-Hopfian groups
Co-Hopfian groups are in some sense weakly dual to Hopfian groups but it does not seem easy to deduce results about co-Hopfian groups directly from those known for Hopfian groups -and vice versa. However given the similarity in the statement of results, we omit some proofs in this subsection, referring the reader to the second author's doctoral thesis [12]. Some results in this subsection are previously appeared in Li [15]. Proposition 4.7. (see [12]). If A, B are co-Hopfian groups and ϕ = (︁ α δ β )︁ represents an injective endomorphism

then ϕ is invertible if one of α, β is injective.
For later reference we note: Corollary 4.8. (see [12]). If A, B are co-Hopfian groups and ϕ = (︁ α δ β )︁ represents an injective endomorphism of G = A ⊕ B, then ϕ is invertible if one of δ, is zero.
We can deduce the following: Proposition 4.9. (see [15]). Suppose that A is a co-Hopfian group and B is a cyclic p-group generated by the element b, then the direct sum G = A ⊕ B is again co-Hopfian.
)︁ be a monomorphism on G = A ⊕ B, β is an integer. We divide the proof into two cases: i) At least one of α, β1 B is injective. In this case, by Proposition 4.7, ϕ is invertible. ii) Both α, β1 B are not injective. Suppose the order of b is o(b) = p n . Clearly p divides β. Since ϕ is injective, we claim Kerα Because of the injectivity of ϕ, we have x = 0 as required. As . Similarly, we have the necessary condition Ker(α + δ ) ∩ Ker( + β ) = {0}.
By induction, we have: Theorem 4.10. (see [15]). Suppose that A is a co-Hopfian group and B is a finite group, then the direct sum G = A ⊕ B is again co-Hopfian.
Note that the direct sum of a co-Hopfian group and a finitely generated group need not be co-Hopfianconsider the group Q ⊕ Z! However, the direct sum of a co-Hopfian group and a finitely co-generated group, does have the desired property. It is worthwhile to note that finitely cogenerated groups are co-Hopfian.
Theorem 4.11. (see [15]). Suppose that A is a co-Hopfian group and B is a finitely co-generated Abelian group, then the direct sum G = A ⊕ B is again co-Hopfian.
Proof. Write A as a direct sum of a reduced subgroup R and a divisible subgroup D 1 , and each summand is therefore a co-Hopfian group; thus the divisible group D 1 is a direct sum of finitely many copies of the group Z(q ∞ ) for various (possibly infinitely many) primes q. The group B is the direct sum of a finite group F and finitely many copies of Z(q ∞ ) for finitely many primes q; call this latter group D 2 . Since, for any fixed prime q, the direct sum of finitely many copies of Z(q ∞ ) is co-Hopfian, it follows from Proposition 4.1 that D 1 ⊕ D 2 is co-Hopfian. Furthermore, the group R ⊕ F is co-Hopfian by Theorem 4.10. Another application of Proposition 4.1 yields the desired result that A ⊕ B is co-Hopfian.