V. S. ZHURAVSKII

ON THE QUESTION OF THE GROUP OF ABELIAN EXTENSIONS OF ABELIAN GROUPS

(Presented by Academician P. S. Aleksandrov on 26 IV 1960)

In the present paper we consider the construction of the group (\operatorname{Ext}(F,T)) of abelian extensions of an abelian group (T) by means of an abelian group (F).

In proving the theorems we shall use the following assertion:

Let (A), (B), and (C) be abelian groups such that the sequence

[
0 \to A \to B \to C \to 0
]

is exact. Then, for any abelian group (G), the sequences

[
0 \to \operatorname{Hom}(G,A) \to \operatorname{Hom}(G,B) \to \operatorname{Hom}(G,C) \to \operatorname{Ext}(G,A) \to
]
[
\to \operatorname{Ext}(G,B) \to \operatorname{Ext}(G,C) \to 0;
\tag{1}
]

[
0 \to \operatorname{Hom}(C,G) \to \operatorname{Hom}(B,G) \to \operatorname{Hom}(A,G) \to
]
[
\to \operatorname{Ext}(C,G) \to \operatorname{Ext}(B,G) \to \operatorname{Ext}(A,G) \to 0.
\tag{2}
]

Theorem 1. Let (F) be a torsion-free abelian group of finite rank (r); let (f_1, f_2, \ldots, f_r) be a system of linearly independent elements of the group (F), and denote (\sum {f_i}=C), (i=1,2,\ldots,r); finally, let (T) be a periodic abelian group. Then the group (\operatorname{Ext}(F,T)) is isomorphic to the quotient group of the group (\operatorname{Ext}(F/C,T)) by its periodic part (V):

[
\operatorname{Ext}(F,T) \cong \operatorname{Ext}(F/C,T)/V.
]

Indeed, since (\operatorname{Hom}(C,T) \cong T \times r) and (\operatorname{Ext}(C,T)=0), the application of the exact sequence (2) to the group (T) and to the exact sequence

[
0 \to C \xrightarrow{\alpha} F \xrightarrow{\beta} F/C \to 0
]

gives the exact sequence

[
T \times r \xrightarrow{w} \operatorname{Ext}(F/C,T) \xrightarrow{\beta^*} \operatorname{Ext}(F,T) \to 0.
]

The group (T \times r) is periodic, the homomorphism (w) maps it into the periodic part (V) of the group (\operatorname{Ext}(F/C,T)), and since (\beta^*) is an epimorphism and (\operatorname{Ext}(F,T)) is a torsion-free group, the homomorphism (w) maps (T \times r) onto the group (V), which is equivalent to the assertion of Theorem 1.

Corollary. Let (T) be a periodic abelian group and let (H) be a torsion-free abelian group of rank one, containing an element (h), in whose characteristic

[
\chi(h)=(k_1,k_2,\ldots,k_i,\ldots)
]

all numbers (k_i) are finite. Then (\operatorname{Ext}(H,T)) is isomorphic to the quotient group of the complete direct sum (U=\sum_{p_i \in P}^{*} T_{p_i}/p_i^{k_i}T_{p_i}) by its periodic part

[
V=\sum_{p_i\in P} T_{p_i}/p_i^{k_i}T_{p_i}, \quad \text{where } T_{p_i}\text{ is the }p_i\text{-primary component of the group }T
]
with respect to the prime number (p_i), and (p_i) runs through the entire set (P) of prime numbers,

[
\operatorname{Ext}(H,T)\cong
\left(\sum_{p_i\in P}^{*} T_{p_i}/p_i^{k_i}T_{p_i}\right)\bigg/
\left(\dot{\sum}{p_i\in P} T\right).}/p_i^{k_i}T_{p_i
]

Thus, denoting by (C) the cyclic subgroup of the group (H) generated by the element (h), we obtain that

[
H/C=\dot{\sum}_{p_i\in P} C(p_i^{k_i}),
]

where (C(p_i^{k_i})) is the cyclic group of order (p_i^{k_i}). Then

[
\operatorname{Ext}(H/C,T)\cong
\sum_{p_i\in P}^{}\operatorname{Ext}(C(p_i^{k_i}),T)
\cong \sum_{p_i\in P}^{} T/p_i^{k_i}T
\cong \sum_{p_i\in P}^{*} T_{p_i}/p_i^{k_i}T_{p_i}.
]

The periodic part (V) of the group (\operatorname{Ext}(H/C,T)) is isomorphic to

[
\dot{\sum}{p_i\in P} T.}/p_i^{k_i}T_{p_i
]

In particular, if (F) is a complete torsion-free group, then a stronger theorem holds.

Theorem 2. Let (F) be a complete torsion-free abelian group of rank (r) ((r) a finite or infinite cardinal number), and let (T) be any reduced abelian group. Then

[
\operatorname{Ext}(F,T)\cong \sum_{r}^{*}(\operatorname{Ext}(R/Z,T)/T),
]

where (R) is the additive group of all rational numbers, (Z) is its subgroup of all integers, and (\sum_{r}^{*}) denotes the complete direct sum of (r) copies.

Indeed, (F=R\times r), (\operatorname{Hom}(R,T)=0), (\operatorname{Hom}(Z,T)\cong T), and (\operatorname{Ext}(Z,T)=0). For the exact sequence

[
0\to Z\to R\to R/Z\to 0,
]

by (2) we obtain the exact sequence

[
0\to T \xrightarrow{\alpha} \operatorname{Ext}(R/Z,T)\to \operatorname{Ext}(R,T)\to 0.
]

Identifying the group (T) with its image (T\alpha), we obtain

[
\operatorname{Ext}(R,T)\cong \operatorname{Ext}(R/Z,T)/T;
]

therefore

[
\operatorname{Ext}(F,T)=\operatorname{Ext}(R\times r,T)
\cong \sum_{r}^{}\operatorname{Ext}(R,T)
\cong \sum_{r}^{}(\operatorname{Ext}(R/Z,T)/T).
]

Lemma. Let (F) and (T) be torsion-free abelian groups, and suppose that the type of every element (f\ne 0) of (F) is greater than the type of every element (t\ne 0) of (T). Then (\operatorname{Hom}(F,T)=0).

Let (\alpha\in \operatorname{Hom}(F,T)), (f\in F), and (f\alpha=t\in T). Comparing the types of the elements (f) and (t) for any (f\in F), we see that (\alpha=0).

Theorem 3. Let the torsion-free abelian groups (F) and (T) satisfy the conditions:

1) (F) is a group of finite rank (r);

2) the type of every nonzero element of (F) is greater than the type of every nonzero element of (T).

If (F_1) is a pure subgroup of rank one of the group (F), then

[
\operatorname{Ext}(F,T)=\operatorname{Ext}(F/F_1,T)+\operatorname{Ext}(F_1,T).
]

Proof. According to the lemma, (\operatorname{Hom}(F/F_1,T)=\operatorname{Hom}(F,T)=\operatorname{Hom}(F_1,T)=0). Therefore, for the exact sequence

[
0\to F_1\to F\to F/F_1\to 0
]

we obtain the exact sequence

[
0\to \operatorname{Ext}(F/F_1,T)\to \operatorname{Ext}(F,T)\to \operatorname{Ext}(F_1,T)\to 0.
]

Now the validity of the theorem follows from the fact that, according to Nunke’s theorem, the group (\operatorname{Ext}(F/F_1,T)) is complete ((^2)).

It is known that every group (H) of rank one is isomorphic to some subgroup of the additive group (R) of rational numbers; we shall identify (H) with this subgroup; by (Z) we denote its subgroup of all integers.

Theorem 4. Let (H) be a group of rank one, and suppose that in the characteristic

[
\chi(1)=(k_1,k_2,\ldots,k_i,\ldots)
]

of the number (1\in H) all numbers (k_i) are finite. Further, let (T) be any torsion-free group. Denote (T'=\bigcap_{p_i\in P} p_i^{k_i}T). Then

[
\operatorname{Ext}(H,T)\cong (\Sigma^{*}T/p_i^{k_i}T)/(T/T').
]

Proof. It is easy to verify that (\operatorname{Hom}(H/Z,T)=0), (\operatorname{Ext}(Z,T)=0), (\operatorname{Hom}(H,T)\cong T'), (\operatorname{Hom}(Z,T)\cong T), and

[
\operatorname{Ext}(H/Z,T)\cong \sum_{p_i\in P}^{*} T/p_i^{k_i}T.
]

Then for the exact sequence

[
0\to Z\to H\to H/Z\to 0
]

there is the exact sequence

[
0\to T'\to T\to \sum_{p_i\in P}^{*} T/p_i^{k_i}T \xrightarrow{\beta} \operatorname{Ext}(H,T)\to 0.
\tag{3}
]

The kernel of the epimorphism (\beta) is isomorphic to the group (T/T'); therefore sequence (3) may be written as the exact sequence

[
0\to T/T' \xrightarrow{\alpha} \sum_{p_i\in P}^{*} T/p_i^{k_i}T \xrightarrow{\beta} \operatorname{Ext}(H,T)\to 0.
]

Identifying the kernel of the epimorphism (\beta) with the group (T/T'), we thereby prove Theorem 4.

Remark. It can be shown that the elements of the kernel ((T/T')\alpha) of the epimorphism (\beta) are precisely those elements of the group

[
\sum_{p_i\in P}^{*} T/p_i^{k_i}T,
]

which can be written in the form of a sequence

[
(t+p_1^{k_1}T,\ t+p_2^{k_2}T,\ \ldots,\ t+p_i^{k_i}T,\ldots)=(\bar t,\bar t,\ldots,\bar t,\ldots),\qquad t\in T.
]

These elements form a subgroup (A) of the group (\sum_{p_i\in P}^{*}T/p_i^{k_i}T), and ((T/T')\alpha=A).

Without particular difficulty, the following corollaries of Theorem 4 can be proved.

Corollary 1. If the group (T) contains elements of type smaller than the type of the group (H), then (\operatorname{Ext}(H,T)) is a mixed group.

Corollary 2. Suppose that the type of every element of the group (T) is not smaller than the type of the group (H). Then the group (\operatorname{Ext}(H,T)) is torsion-free, and

[
\operatorname{Ext}(H,T)\cong
\left(\sum_{p_i\in P}^{}T/p_i^{k_i}T\right)
\bigg/
\left(\sum_{p_i\in P}^{}T_i/p_i^{k_i}T_i\right).
]

Corollary 3. Let the groups (H) and (T) satisfy the condition of Theorem 4. In this case the group (\operatorname{Ext}(H,T)) is equal to zero if and only if at least one of the following conditions is fulfilled:

1) in the characteristic (\chi(1)=(k_1,k_2,\ldots,k_i,\ldots)) almost all the numbers (k_i) are equal to (0);

2) the group (T) is a (p)-complete group for almost all those primes for which in (\chi(1)) the numbers (k_i\ne 0).

It is known that the concept of a basic subgroup of a given primary abelian group, introduced by L. Ya. Kulikov, has proved very fruitful in the theory of abelian groups. A. G. Kurosh posed the question:

Will the construction of the group (\operatorname{Ext}(F,T)) be simplified if one introduces into consideration a basic subgroup of the group (F) or (T)?

Along this path the following results have been obtained.

Theorem 5. Let (F) be a group primary with respect to the prime (p), and let (B) be its basic subgroup. If the abelian group (T) satisfies the conditions:

1) (T[p]=0);

2) (pT=T),

then

[
\operatorname{Ext}(F,T)\cong \operatorname{Ext}(F/B,T).
]

Proof. It is easy to see that
[
\operatorname{Hom}(F/B,T)=\operatorname{Hom}(F,T)=\operatorname{Hom}(B,T)=\operatorname{Ext}(B,T)=0.
]
And then the application of the exact sequence (2) to the group (T) and the exact sequence

[
0\to B\to F\to F/B\to 0
]

proves Theorem 5.

Theorem 6. Let (T) be a periodic abelian group whose primary components (T_p) do not contain elements of infinite height, and let (B) be the direct sum of the basic subgroups of the primary components of the group (T),

[
B=\sum_{p\in M} B_p,
]

and let (F) be such a torsion-free group that (pF=F) for all primes (p) for which the primary components (T_p\ne 0). Then the exact sequence

[
0\to \operatorname{Hom}(F,T/B)\to \operatorname{Ext}(F,B)\to \operatorname{Ext}(F,T)\to 0
]

holds.

To the group (F) and the exact sequence

[
0\to B\to T\to T/B\to 0
]

the exact sequence (1) is applied, and it is proved that
[
\operatorname{Hom}(F,B)=\operatorname{Hom}(F,T)=\operatorname{Ext}(F,T/B)=0.
]

The author expresses his deep gratitude to A. G. Kurosh for his guidance in carrying out this work.

Received
22 IV 1960

REFERENCES

(^{1}) A. G. Kurosh, Theory of Groups, Moscow—Leningrad, 1953.
(^{2}) R. Baer, Math. Ann., 135, no. 3, 219 (1958).
(^{3}) S. Mac Lane, Séminaire Dubreil et Pisot, 7 mai 1956.