can someone help me out with this question? LCA stands for Locally compact Hausdorff abelian group. The question is posted in the attached image
Let $A$ and $B$ be LCA-groups and $H$ a (not necessarily closed) finitely generated subgroup of $A$. If $f$ is a continuous homomorphism of $A$ into $B$ such that the kernel of $f$ lies wholly in $H$ and its topologically isomorphic to $Z^n$, for some $n\ge1$, and such that $f(H)$ contains a subgroup topologically isomorphic to $Z$, show that $H$ contains a subgroup topologically isomorphic to $Z^{n+1}$.
It seems the following.
We don’t have much choice to find this subgroup. Let $Z$ be a subgroup of the group $f(H)$, topologically isomorphic to $\Bbb Z$. Put $D=\ker f$ and $C=f^{-1}(Z)$. Since the map $f$ is continuous, the group $C$ is a closed subgroup of a locally compact group $G$. So $C$ is a locally compact group too. By Theorem 2.3 from [HC], every locally compact Hausdorff space is Baire. So the space $C$ is Baire. Since $C$ is a countable Baire topological group, it is discrete. It is well-known, that if a quotient group $C/D$ of an abelian group $C$ by its subgroup $D$ is a free group then $D$ is a direct summand of the group $C$ (see, for instance, [Kur, the end of $\S 19$]). Since the group $C/D$ is (algebraically) isomorphic to $\Bbb Z$, and the group $D$ is isomorphic to the group $\Bbb Z^{n}$, the group $C$ is isomorphic to he group $\Bbb Z^{n+1}$. Since the group $C$ is discrete, it is topologically isomorphic to the group $\Bbb Z^{n+1}$.
Remark 1. We did not use the fact that the group $H$ is finitely generated.
Remark 2. It is a following useful theorem (see, [Pon, Th. 13]). Let $G$ be a locally compact $\sigma$-compact topological group and $N_1,\dots, N_k$ be closed normal subgroups of the group $G$. If $G$ algebraically splits into a direct product of its subgroups $N_1,\dots, N_k$ then the topological group $G$ splits into a direct product of its subgroups $N_1,\dots, N_k$ too.
References
[HC] R. C. Haworth, R. C. McCoy, Baire spaces, Warszawa, Panstwowe Wydawnictwo Naukowe, 1977.
[Kur] A. G. Kurosh, Group theory, 3nd ed., Nauka, Moskow, 1967. (in Russian)
[Pon] L. S. Pontrjagin, Continuous groups, 2nd ed., M., (1954) (in Russian).