Theory and Applications of Categories, Vol. 5, No. 4, pp. 81–90.

A USEFUL CATEGORY FOR MIXED ABELIAN GROUPS.

GRIGORE CĂLUGĂREANU

Transmitted by F. William Lawvere

ABSTRACT. All the useful categories in the study of the mixed abelian groups (e.g. Warf and Walk) ignore the torsion. We introduce a new category denoted A which ignores the torsion-freeness and could characterize some classes of nonsplitting mixed groups with the aid of Walk.

1. Introduction

The categories Warf , first introduced as H in [7] and Walk, first introduced as C in [2] have useful applications in the theory of the mixed abelian groups. In what follows we introduce the category A whose objects are all the abelian groups (i.e. Ob(A) =Ob(Ab)) and whose morphisms, are A(G, H) = Ab(G, H)/J(G, H) where

J(G, H) = {f : G → H|T (G) ≤ ker(f)} ,

for G, H ∈ Ob(A), study its categorical properties and establish connections with the above mentioned category Walk. Finally, some results that justify the utility of this category are given.

Needless to say, all the groups considered will be abelian.

2. The categorical structure

For two groups G and H, we consider on the abelian group Ab(G, H) the binary relation

ρ_{G,H} defined by (f, g) ∈ ρ_{G,H} ⇔ T (G) ⊆ ker(f − g) where G \begin{matrix}f \\ ⟶ \\ ⟶ \\ g \end{matrix} H.

2.1. Lemma. For α, β ∈ Ab(G, H) the inclusion kerα ∩ kerβ ⊆ ker(α + β) holds.

2.2. Proposition. The relation ρ_{G,H} is a congruence relation.

Proof. Indeed, using 2.1 two times, the relation ρ_{G,H} is :

• reflexive T (G) ⊆ G = ker(0) = ker(f − f) ⇒ (f, f) ∈ ρ_{G,H}, ∀f ∈ Ab(G, H)

• symmetric (f, g) ∈ ρ_{G,H} ⇒ T (G) ⊆ ker(f − g) = ker(g − f) ⇒ (g, f) ∈ ρ_{G,H} • transitive (f, g), (g, h) ∈ ρ_{G,H} ⇒ T (G) ⊆ ker(f − g), ker(g − h) ⇒ T (G) ⊆ ker(f −

Received by the editors 1998 April 21 and, in revised form, 1999 January 27.

Published on 1999 March 9.

1991 Mathematics Subject Classification: 20K21, 18E05.

Key words and phrases: additive category, quotient category, splitting mixed Abelian groups.

\begin{matrix}c \\ ◯ \end{matrix} Grigore Călugăreanu 1999. Permission to copy for private use granted.

g) ∩ ker(g − h) ⊆ ker((f − g) + (g − h)) = ker(f − h) ⇒ (f, h) ∈ ρ_{G,H}.

Moreover, if (f, g), (f_{1}, g_{1}) ∈ ρ_{G,H} then T (G) ⊆ ker(f − g) ∩ ker(f_{1} − g_{1}) ⇒ T (G) ⊆ ker((f − g) + (f_{1} − g_{1})) = ker((f + f_{1}) − (g + g_{1})) ⇒((f + f_{1}), (g + g_{1})) ∈ ρ_{G,H}.

There is a well-known order isomorphism between congruences and subgroups: J(G, H) = ρ_{G,H} ⟨0⟩ = {f ∈ Ab(G, H)|T (G) ⊆ ker(f)} is the corresponding subgroup.

Elementary: T (G) ⊆ ker(f) ∩ ker(g) ⊆ ker(f ± g), so that

2.3. Remark. For every f, g ∈ J(G, H) also f ± g ∈ J(G, H) holds.

Clearly, if T is a torsion group, J(T, H) = {0} for every group H and so A(T, H) = Ab(T, H).

2.4. Lemma. (a) kerα ⊆ ker(β ∘ α); (b) For G ⟶^{α} H ⟶ K^{β}, T (H) ⊆ kerβ ⇒ T (G) ⊆ ker(β ∘ α).

Proof. (b) Indeed, x ∈ T (G) ⇒ α(x) ∈ T (H) ⊆ kerβ ⇒ x ∈ ker(β ∘ α).

2.5. Proposition. The relations {ρ_{G,H}|G, H ∈ Ob(Ab)} are compatible with composition.

Proof. Indeed, using Lemma 2.4 (a) and (b), one has: (f, g) ∈ ρ_{G,H}, (f,′ g′) ∈ ρ_{H,K} ⇒ ker(f ∘ f′ − g ∘ g′) = ker((f′ ∘ (f − g) + (f′ − g′) ∘ g) ⊇

ker(f′ ∘ (f − g)) ∩ ker((f′ − g′) ∘ g) ⊇^{2.4} ker(f − g) ∩ T (G) ⊇ T (G) ⇒

(f′ ∘ f, g′ ∘ g) ∈ ρ_{G,K}.

Hence, we define the category A, as a quotient category of Ab whose objects are all the abelian groups (i.e. Ob(A) = Ob(Ab)) and whose morphisms, for each two groups G, H are given by A(G, H) = Ab(G, H)/J(G, H) (or Ab(G, H)/ρ_{G,H}). We shall denote the classes \overline{f} = f + J(G, H) in A(G, H). The composition in A is well-defined according to the above Proposition and 1_{G} + J(G, G) is the identity morphism. Associativity and bilinearity are easily verified (using 2.2) so that

2.6. Theorem. A is an additive category.

For the following elementary results we use the notation: if f : G → H then f|_{T (G)} : T (G) → H and f̃|_{T (G)} : T (G) → T (H) (because im(f|_{T (G)}) ⊆ T (H)).

2.7. Proposition. (a) f + J(G, G) is the identity in A(G, G) iff f|_{T (G)} : T (G) → G is the inclusion (i.e. f fixes the finite order elements);

(b) if f|_{T (G)} or f̃|_{T (G)} is a monomorphism in Ab then f +J(G, H) is a monomorphism in A(G, H);

(c) if f̃|_{T (G)} is an epimorphism in Ab then f +J(G, H) is an epimorphism in A(G, H). If H splits over T (H), the converse also holds.

Proof. Clearly, the equality in A is characterized as follows: \overline{f} = f + J(G, H) = \overline{g} = g + J(G, H) ⇔ f − g ∈ J(G, H) ⇔ T (G) ≤ ker(f − g) ⇔

(f − g)(T (G)) = 0 ⇔ f|_{T (G)} = g|_{T (G)}.

Hence, for (a) it suffices to observe that 1_{G}|_{T (G)} : T (G) → G is the inclusion.

(b) For L \begin{matrix}α \\ ⟶ \\ ⟶ \\ β \end{matrix} G ⟶^{f} H and f|_{T (G)} monic in Ab suppose \overline{f} ∘ \overline{α} = \overline{f} ∘ \overline{β}. Then \overline{f ∘ α} =

\overline{f ∘ β} and f ∘α|_{T (L)} = f ∘β|_{T (L)}. Using α̃|_{T (L)} : T (L) → T (G) (indeed, im(α|_{T (L)}) ⊆ T (G)) and f ∘ α|_{T (L)} = f|_{T (G)} ∘ α̃|_{T (L)} we derive α̃|_{T (L)} = β̃|_{T (L)} or α|_{T (L)} = β|_{T (L)}. Hence \overline{α} = \overline{β}.

(c) For G ⟶^{f} H \begin{matrix}α \\ ⟶ \\ ⟶ \\ β \end{matrix} L and f̃|_{T (G)} epic in Ab suppose \overline{α}∘\overline{f} = \overline{β}∘\overline{f}. Then \overline{α ∘ f} = \overline{β ∘ f}

and α ∘ f|_{T (G)} = β ∘ f|_{T (G)}. As above α ∘ f|_{T (G)} = α|_{T (H)} ∘ f̃|_{T (G)} so that α|_{T (H)} = β|_{T (H)} and \overline{α} = \overline{β}.

If T (H) is a direct summand of H, all homomorphisms σ, τ : T (H) → L extend to

morphisms σ_{1}, τ_{1} : H → L. Now, set T (G) \underrightarrow{_{f|_{T (G)}}} T (H) \begin{matrix}σ \\ ⟶ \\ ⟶ \\ τ \end{matrix} L such that σ ∘ f̃|_{T (G)} =

τ ∘ f̃|_{T (G)}. As before, using any extensions σ_{1}, τ_{1} we derive σ_{1} ∘ f|_{T (G)} = σ_{1}|_{T (H)} ∘ f̃|_{T (G)} = σ ∘ f̃|_{T (G)} = τ_{1} ∘ f|_{T (G)} or \overline{σ_{1} ∘ f} = \overline{τ_{1} ∘ f}. Hence \overline{σ_{1}} = \overline{τ_{1}} or σ_{1}|_{T (H)} = τ_{1}|_{T (H)} and σ = τ.

2.8. Remark. The groups G such that for every group H, each homomorphism σ : T (G) → H extends to a homomorphism σ_{1} : G → H are exactly the splitting ones.

Indeed, for H = T (G) and σ = 1_{T (G)} there is an extension u : G → T (G) such that u ∘ i = 1_{T (G)}, where i : T (G) → G is the inclusion.

2.9. Remark. A is not balanced and so, not normal nor conormal.

Proof. Consider the inclusion i : T (G) → G of the torsion part of a nonsplitting mixed group G such that T (G) is no epimorphic image of G (e.g. ∏_{p∈P} Z(p) ∉ M_{1} (see

[9])). According to the proposition above \overline{i} ∈ A(T (G), G) is a monomorphism and an epimorphism but not an isomorphism in A. Indeed, if \overline{i} should be an isomorphism in A there would exist a morphism π : G → T (G) in Ab such that \overline{π}∘ \overline{i} = \overline{1}_{T (G)}, \overline{i} ∘ \overline{π} = \overline{1}_{G} in A. Hence π|_{T (G)} = 1_{T (G)} and so π would be an epimorphism.

2.10. Theorem. In A the torsionfree groups are exactly the zero objects. In particular, all the torsionfree groups are A-isomorphic.

Proof. A group G is an initial object in A iff Ab(G, H) = J(G, H) holds for each group H. Hence G is initial iff T (G) ≤ ker(f) holds for each group H and each homomorphism f : G → H. Taking f any injective homomorphism we obtain T (G) = 0. Conversely, if T (G) = 0 surely T (G) ≤ ker(f) holds for every H and every f. Hence J(G, H) = Ab(G, H) and A(G, H) = Ab(G, H)/Ab(G, H) = \{\overline{0}\} .

Further, G is a terminal object in A iff Ab(H, G) = J(H, G) holds for each group H. Hence G is terminal iff T (H) ≤ ker(f) holds for each group H and each homomorphism f : H → G. Taking H = G, f = 1_{G} we obtain T (G) = 0. Conversely, T (G) = 0 implies T (H) ≤ ker(f) for each group H and each homomorphism f : H → G. Indeed, f(T (H)) ⊆ T (G) implies f(T (H)) = 0 and so T (H) ≤ ker(f).

Hence the zero objects in A are the torsionfree groups.

2.11. Theorem. A has cokernels.

Proof. Finally, for f + J(G, H) ∈ A(G, H), if p : H → \overline{H} = H/(f(T (G)) denotes the canonical projection, we verify that p + J(H, \overline{H}) = coker(f).

First, p∘f ∈ J(G, \overline{H}). Indeed, T (G) ≤ ker(p∘f) ⇔ (p∘f)(T (G)) = 0 ⇔ p(f(T (G)) = 0, which clearly holds. Next, if the following diagram commutes

L
[figure: 0]
[figure: G; f; H; g]
[figure: 0]
\overline{H}p_{}

there is a unique homomorphism h : \overline{H} → L such that the following triangle commutes

L
[figure: h]
[figure: H; g; p; \overline{H}]

Indeed, g ∘ f = 0 in A iff g ∘ f ∈ J(G, L). This is consequently equivalent to T (G) ≤ ker(g∘f) ⇔ f(T (G)) ≤ ker(g) and so, to ker(p) ≤ ker(g). Hence a unique homomorphism h : \overline{H} → L exists such that the above triangle commutes.

2.12. Remark. For each G, H the group A(G, H) can be identified with a subgroup of Ab(T (G), T (H)).

Indeed, first observe that J(G, H) can be identified with Ab(G/T (G), H). Indeed, T (G) ≤ ker(f) implies that there is a unique homomorphism \underline{f} : G/T (G) → H with f = p_{T (G)} ∘ \underline{f}. Next, use the left exactness of the contravariant functor Ab(−, H) for the short exact sequence 0 → T (G) → G → G/T (G) → 0. We obtain the exact sequence 0 → Ab(G/T (G), H) → Ab(G, H) →^{t} Ab(T (G), H) and then A(G, H) = Ab(G, H)/J(G, H) ≅ Ab(G, H)/Ab(G/T (G), H) ≅

Ab(G, H)/ker(t) ≅ im(t), which can be identified with a subgroup of

Ab(T (G), T (H)).

2.13. Theorem. The category A has products.

Proof. Let {f_{i} + J(G, G_{i}) : G → G_{i}} be a family of morphisms in A and {p_{j} : ∏_{i∈I} G_{i} →

G_{j}, ∀j ∈ I} the canonical projections for the direct product (from Ab). Clearly there is a unique f : G → ∏_{i∈I} G_{i} such that f_{i} = p_{i} ∘ f. One easily checks that ∀i ∈ I : g_{i} ∈

f_{i} + J(G, G_{i}), g_{i} = p_{i} ∘ g implies g ∈ f + J(G, ∏_{i∈I} G_{i}).

Indeed, T (G) ≤ ker(g_{i} − f_{i}), ∀i ∈ I ⇒ T (G) ≤ ker(g − f) because ker(g − f) =

⋂_{i∈I} ker(g_{i} − f_{i}).

Clearly, there is a unique factorization f_{i} + J(G, G_{i}) = (p_{i} + J(∏_{i∈I} G_{i}, G_{j})) ∘ (f +

J(G, ∏_{i∈I} G_{i})).

Notice that in A there are finite direct sums (products) defined as usually in Ab (as objects). Moreover

2.14. Theorem. A has infinite coproducts (direct sums).

Proof. Let {f_{i} + J(G_{i}, G) : G_{i} → G} be a family of morphisms in A. The proof is similar to the previous one: it reduces to the inclusion ⨁_{i∈I} ker(f_{i}) ≤ ker(f) and the equality

T (⨁_{i∈I} G_{i}) = ⨁_{i∈I} T (G_{i}), where f_{i} = f ∘ q_{i} gives the unique decomposition with {q_{j} : G_{j} → ⨁_{i∈I} G_{i}, ∀j ∈ I} the canonical injections into the coproduct (direct sum).

2.15. Remark. A does not have kernels.

Indeed, for a morphism \overline{f} ∈ A(G, H), (T (G) ∩ ker(f), \overline{incl}) must be the kernel in A. But this is not the case in general.

2.16. Theorem. Two groups G and H are isomorphic in A iff there are two torsion-free groups U and V such that G ⊕ U ≅ H ⊕ V.

Proof. The condition is sufficient: first, notice that if U is torsion-free, the canonical projection p_{G} : G ⊕ U → G, respectively injection e_{G} : G → G ⊕ U have mutually inverse classes in A. Indeed, p_{G} ∘ e_{G} = 1_{G} implies \overline{p_{G}} ∘ \overline{e_{G}} = \overline{1_{G}} in A and conversely, (e_{G} ∘ p_{G}, 1_{G⊕U}) ∈ ρ_{G⊕U,G⊕U}, this being justified as follows: ker(1_{G⊕U} − e_{G} ∘ p_{G}) =

ker (e_{U} ∘ p_{U}) = G ≥ T (G) = T (G⊕U). Then G ≅^{A} G⊕U and one uses also G⊕U ≅ H ⊕V

and similarly H ⊕ V ≅^{A} H.

The condition is also necessary: suppose G ≅^{A} H, that is, there are homomorphisms f : G → H and g : H → G such that f ∘g−1_{H} = s ∈ J(H, H) and g∘f −1_{G} = t ∈ J(G, G).

First observe that the restrictions f|_{T (G)} : T (G) → T (H), g|_{T (H)} : T (H) → T (G) are mutually inverses in Ab (indeed, e.g. f|_{T (G)} ∘ g|_{T (H)}(h) = 1_{H}(h) + s(h) = h = 1_{T (H)}(h), ∀h ∈ T (H) using T (H) ≤ ker(s)). Define P as the pushout in the following commutative diagram

[figure: 0; T (G); ≅^{f|_{T (G)}}; T (H); in; H; pr; H/T (H); 0]
[figure: g]
[figure: 0; G; in; P; H/T (H); ‖; 0]

A well-known exercise from abelian category theory shows that the bottom line is also exact . As g : H → G renders the upper triangle commutative, the bottom line splits and so P ≅ G ⊕ H/T (H). Using the 3 × 3−lemma the same pushout may be used

0  0  0

[figure: 0; T (G); ≅^{f|_{T (G)}}; T (H); in; H; pr; H/T (H); 0]

[figure: 0; G; in; P; H/T (H); ‖; 0]

[figure: 0; G/T (G); pr; =; G/T (G); 0; 0]

[figure: 0; 0; 0]

in order to prove that P ≅ H ⊕ G/T (G). Hence G ⊕ H/T (H) ≅ H ⊕ G/T (G) with torsion-free groups H/T (H) and G/T (G).

Similarly to [7], one can write down an isomorphism in terms of the given functions f and g : indeed

G ⊕ H/T (H) ⟶ H ⊕ G/T (G)

(x, y + T (H)) ⟼ (f(x − g(y)) + y, x − g(y) + T (G))

and

H ⊕ G/T (G) ⟶ G ⊕ H/T (H)

(y, x + T (G)) ⟼ (g(y − f(x)) + x, y − f(x) + T (H))

define group morphisms that are inverses of one another.

As in [3], one can also prove the above result by G ⊕ H/ker(s) ≅ H ⊕ G/ker(t).

Then

2.17. Corollary. G ≅^{A} H and G ≅^{Walk} H iff there are torsion groups S, T and torsionfree groups U, V such that G ⊕ S ≅ H ⊕ T and G ⊕ U ≅ H ⊕ V.

3. A full embedding

As in [4] there is a natural embedding of To, the full subcategory of Ab which consists of all the torsion abelian groups, into A.

3.1. Theorem. The functor I : To → A, defined by I(T) = T on objects and I_{T S} : To(T, S) → A(T, S), I_{T S}(f) = f + J(T, S) = {f} on morphisms, is a full embedding.

Proof. Indeed, as we already have noticed for any T ∈ Ob(To), J(T, G) = {0} and hence A(T, S) = {{f} |f ∈ Ab(T, S)} .

3.2. Theorem. I has an adjoint (to the right): K : A→ To, defined K(G) = T (G) on objects and K_{GH}(\overline{f}) = f̃|_{T (G)} on morphisms.

Proof. First of all, notice that K is well-defined (see the characterization of the equality of the morphisms in A : for each G, H ∈ A we can consider K_{GH}(\overline{f}) = f̃|_{T (G)} because \overline{f} = \overline{g} ⇔ f|_{T (G)} = g|_{T (G)} ⇔ f̃|_{T (G)} = g̃|_{T (G)}). Next, for the adjoint situation the unit η : 1_{To} → K ∘ I is trivially given by the identity 1_{T} : T → K(I(T)) = T for each T ∈ Ob(To), and the counit ε : I ∘ K → 1_{A} is given by the inclusion ε_{G} : I(K(G)) = T (G) → G for each G ∈ Ob(A), all these being natural transformations. Moreover, one easily verifies K →^{η·K} K ∘I ∘K →^{K·ε} K = 1_{K} and I →^{I·η} I ∘K ∘I →^{ε·I} I = 1_{I}. [Another proof: one verifies the natural equivalence of abelian group-valued bifunctors α_{T,G} : A(I(T), G) → To(T, K(G)), ∀T ∈ Ob(To), ∀G ∈ Ob(A)].

3.3. Corollary. K is a limit preserving monofunctor and I is a colimit preserving epifunctor.

Indeed, this is a known property of functors which admit an adjoint to the right.

3.4. Corollary. I also reflects colimits.

Use the dual of Ex. 27H(c), p.204,[1].

3.5. Remark. I is not an equivalence of categories.

Indeed, I is an equivalence ⇔ I is dense (representative) ⇔ ∀G ∈ Ob(Ab) =

Ob(A), ∃T ∈ Ob(To) : T = I(T) ≅^{A} G.

This is also equivalent with the existence of two torsion-free groups U, V such that G ⊕ U ≅ T ⊕ V. Taking the torsion parts (we apply the functor \overline{T} : Ab → Ab, \overline{T} (G) = T (G), ∀G ∈ Ab) of these groups we observe that T (G) ≅ T.

Hence, I is an equivalence ⇔ ∀G ∈ Ob(Ab), ∃U, V torsion-free groups : G ⊕ U ≅ T (G) ⊕ V.

For splitting mixed groups this last condition holds when

a) G is torsion: obviously U = V = 0;

b) G is torsion-free: obviously U = T (G) = 0, V = G;

c) G is splitting mixed, say G = T ⊕ F : obviously U = 0, V = F.

So in order to prove that I is not an equivalence an example of non-splitting mixed group M such that ∀U, V torsion-free groups, M ⊕ U ≅ T̸ (M) ⊕ V suffices.

4. Relations with Walk

The category Walk was also defined as a quotient category of Ab by Ob(Walk) = Ob(Ab) and Walk(G, H) = Ab(G, H)/I(G, H) where

I(G, H) = {f ∈ Ab(G, H)|im(f) ⊆ T (H)} or, similarly with A, with the aid of a congruence relation ω_{G,H} defined by (f, g) ∈ ω_{G,H} ⇔ im(f − g) ⊆ T (H).

Notice that f ∈ I(G, H) ⇔ G = f^{−1} (T (H)).

4.1. Remark. I(G, H) ∩ J(G, H) can be identified with Ab (G/T (G), T (H)) .

Indeed, f ∈ I(G, H) ∩ J(G, H) iff there is a unique f_{1} ∈ Ab (G/T (G), T (H)) such that f = p_{T (G)} ∘ f_{1} ∘ in, the inclusion in : T (H) → H. The situation is described in the following canonical commutative diagram

G  \overline{f ^{}}H
[figure: p_{T (G)}; in]
[figure: G/T (G); pr; G/kerf; f_{0}; imf; incl; T (H)]

with f_{1} = incl ∘ f_{0} ∘ pr(as for the converse, im(p_{T (G)} ∘ f_{1} ∘ in) ≤ im(in) = T (H) resp. T (G) = ker(p_{T (G)}) ≤ ker(p_{T (G)} ∘ f_{1} ∘ in)).

4.2. Remark. {f ∈ Ab(G, H)|f^{−1}(T (H)) is a direct summand of G} ≤ I(G, H) + J(G, H).

Proof. Indeed, for any f ∈ Ab(G, H) set S = f^{−1}(T (H)) = {x ∈ G|f(x) ∈ T (H)} , the preimage. Surely, T (G) ≤ S and ker(f) = f^{−1}(0) ≤ S. If S is a direct summand and G = S ⊕ K, consider g ∈ I(G, H), g(s + k) = f(s), ∀s ∈ S, k ∈ K (i.e. img ≤ f(S) ≤ T (H)) and h ∈ J(G, H), h(s + k) = f(k), ∀s ∈ S, k ∈ K (i.e. T (G) ≤ S ≤ kerh). Clearly f = g + h.

A more categorical proof was pointed out by the referee: if G = S ⊕ K and i_{S}, i_{K} respectivelly p_{S}, p_{K} denote the canonical injections respectivelly projections then i_{S} ∘p_{S} + i_{K} ∘ p_{K} = 1_{G} so that f = f ∘ i_{S} ∘ p_{S} + f ∘ i_{K} ∘ p_{K}. Clearly, f ∘ i_{S} ∘ p_{S} ∈ I(G, H) and f ∘ i_{K} ∘ p_{K} ∈ J(G, H).

5. Endomorphism rings in A

In Warf and Walk the endomorphism rings for torsion-free rank one groups are characterized (see [8] and [5]).

If we denote End_{A}(G) = A(G, G) for any group G then

5.1. Theorem. The map α : End_{A}(G) → End_{Ab}(T (G)), α(f + J(G, G)) = f̃|_{T (G)} is a ring embedding. If G splits, this is a ring isomorphism.

Proof. Indeed, g ∈ f + J(G, G)) ⇔ f|_{T (G)} = g|_{T (G)} shows that α is well-defined and injective. The compatibility with addition and composition are immediate. If G splits, the endomorphisms of T (G) extend to the whole G and so α is also surjective.

6. Classification

Walk was constructed as a quotient category of Ab in order to neglect torsion. Similarly, A is a quotient category of Ab which neglects torsion-freeness. It is natural to ask to what extent these two quotient categories characterize classes M of abelian groups.

Using 2.17 we easily get

6.1. Proposition. If G ≅^{Walk} H and G ≅^{A} H then T (G) ≅ T (H) and G/T (G) ≅ H/T (H).

Proof. If there are torsion groups S, T and torsion-free groups U, V such that G ⊕ S ≅ H ⊕ T and G ⊕ U ≅ H ⊕ V then T (G) = T (G ⊕ U) ≅ T (H ⊕ V) = T (H). Further, G/T (G) ≅ \frac{G⊕S}{T (G)⊕S} = \frac{G⊕S}{T (G⊕S)} ≅ \frac{H⊕T}{T (H⊕T)} = \frac{H⊕T}{T (H)⊕T} ≅ H/T (H) the second isomorphism being obtained as G/kerλ ≅ im(λ) for λ = pr ∘ inj : G → G ⊕ S → \frac{G⊕S}{T (G)⊕S} .

6.2. Corollary. G ≅^{Walk} H and G ≅^{A} H characterize the class of all the splitting mixed groups.

Finally, some open problems:

Problem 1. Are the groups G such that T (G) ≅^{A} G exactly the splitting (mixed) groups? Problem 2. Find classes M of abelian groups such that two groups G and H are isomorphic (in M) iff G and H are isomorphic in Walk and in A.

As for this last problem, following definitions from [9], the classes M_{1} and M_{2} of mixed abelian groups could be considered. Recall that

G ∈ M_{1} if T (G) is a homomorphic image of G and G ∈ M_{2} if G/T (G) can be embedded in G.

References

[1] Herrlich H., Strecker G., Category Theory, second edition, Helderman Verlag, Berlin, 1979.

[2] Hunter R., Richman F., Walker E., Warfield modules, Abelian Group Theory, Lecture Notes in Math. 616 (1976), 87-123, Springer Verlag.

[3] Fuchs L., Abelian p-groups and mixed groups, Séminaire de Mathématiques Supérieures, Les Presses de l’Université de Montréal, 1980.

[4] Franzen B., Goldsmith B., On endomorphism algebras of mixed modules, J.London Math.Soc. (2) 31 (1985), 468-472.

[5] Goldsmith B., Zanardo P., The Walker endomorphism algebra of a mixed module, Proc. R. Irish Acad., 93A (1) (1993), 131-136.

[6] Mitchell B., The Theory of Categories, Academic Press, New York and London, 1965.

[7] Warfield R.B., Classification theory on abelian groups,I: balanced projectives, Trans.Amer.Math.Soc., 22 (1976), 33-63.

[8] Warfield R.B., The structure of mixed abelian groups, Abelian Group Theory, Lecture Notes in Math. 616 (1976), 1-38, Springer Verlag.

[9] Weinstein M., Examples of Groups, Polygonal Publishing House, 1977.

Dept. of Mathematics, ”Babeş-Bolyai” University, Kogălniceanu str. 1, 3400 Cluj-Napoca, Romania

Email: calu@math.ubbcluj.ro

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