n H\right|=|H|, \mbox{ for all }n\in {{\mathbb N}}.$$

Decompose $t(G)=\bigoplus_p G_p$ as a direct sum of $p$-groups $G_p$ and let $B_p$ denote a basic subgroup of $G_p$ for each $p$. This in particular means that $B_p$ is a direct sum of cyclic $p$-groups, $$B_p = \bigoplus_{n < \omega} B_{p,n} \mbox{  with }
B_{p,n} \cong \bigoplus_{\beta_{p^n}}{\textbf}Z(p^n)$$ and that $G_p/B_p$ is divisible. Define $\mathcal{D}=\{|B_{p,n}|\colon p^n \in
\mathbb{P}^\uparrow\}$. If $\mathcal{D}$ has no maximum or $\beta_0=\max \mathcal{D}$ is attained at an infinite number of $|B_{p,n}|$’s we stop here. If, otherwise, $\beta_0=\max
\mathcal{D}=|B_{p_1,n_1}|=\ldots=|B_{p_r,n_r}|$ and $|B_{p_j,n_j}|<\beta_0$ for all the remaining $p_j^{n_j}\in
{{\mathbb P}}^\uparrow$ we repeat the process with the set $\mathcal{D}\setminus |B_{p_1,n_1}|$. After a finite number of steps we obtain in this manner a finite collection of cardinals $F\subset
\mathcal{D}$ such that either:

1.  *Case 1:* the supremum $\beta:=\sup\left( \mathcal{D}\setminus F\right)$ is not attained, or

2.  *Case 2:* the supremum $\beta:=\sup \left(\mathcal{D}\setminus F\right)$ is attained infinitely often, i.e., there is an infinite subset $I\subset {{\mathbb P}}^{\uparrow}$ with $|B_{p,n}|=\beta$ for all $p^n \in I$.

Define ${{\mathbb P}}_0^{\uparrow}=\{ p^n \in {{\mathbb P}}^\uparrow \colon  |B_{p,n}|\in
F\}$ (observe that ${{\mathbb P}}_0^\uparrow$ is necessarily finite), and set $\gamma(p_k^{n_k})=|B_{p_k,n_k}|$ if $p_k^{n_k}\in {{\mathbb P}}_0^{\uparrow}$. Since the subgroups $B_{p_k,n_k}$ are bounded pure subgroups, there 