We know that the generalized Prüfer $p$ -group $H_{\omega+1}$ is a group having the following generators and relations respectively
$$X=\{a_0,a_1,a_2,\ldots\},~~~ \{pa_0=0, p^na_n=a_0,~ \text{for all}~ n\geq 1\}.$$
Also, we know that $H_{\omega+1}$ is not an indecomposable gruop (moreover, it is not a direct sum of indecomposable groups).
${\bf Question}.$ Give a non-trivial direct sum decomposition $H_{\omega+1}=A\oplus B$ of $H_{\omega+1}$.