Let $\Gamma$ be a commutative group. Let there be a function $h:\Gamma\to[0,\infty)$ with the following properties.
For every real number $M$, the set $\{P\in\Gamma : h(P)\leq M\}$ is finite.
For every $P_0\in\Gamma$, there is a constant $k_0$ so that $h(P+P_0)\leq2h(P)+k_0$ for all $P\in\Gamma$
There is a constant $k$ so that $h(2P)\geq4h(P)-k$ for all $p\in\Gamma$
Such a function is called a height function. If the subgroup $2\Gamma$ has finite index in $\Gamma$ then existence of a height function implies the group is finitely generated.
One example function is defined on the group of rational points of an elliptic curve over $\Bbb{Q}$ by $h(\frac{m}{n})=max\{|m|,|n|\}$, where $\frac{m}{n}$ is the $x$ coordinate of an element written in lowest terms. This function is used to prove Mordell's theorem that states the group of rational points on an elliptic curve over $\Bbb{Q}$ is finitely generated.
For an arbitrary commutative group $\Gamma$, does there always exist a height function? If yes then what are some examples of such functions?
For an arbitrary commutative group the answer must be no, for example, because condition (3) implies that $$ \{P \in \Gamma: 2P = 0\} \subseteq \{P \in \Gamma: h(P) \leq (h(0)+k)/4\}, $$ and while the latter set must be finite by condition (1), there is no general reason why the former set must be finite. As extreme cases one can consider, for example, that there are infinite commutative groups $\Gamma$ for which $2 \Gamma = \{0\}$, such as infinite direct sums of copies of $\mathbb{Z}/2\mathbb{Z}$.
More generally, if there is a function $h$ on a commutative group $\Gamma$ satisfying condition (3) and $Q \in \Gamma$ has finite order, it is possible to show that $h(Q) \leq k/3$. It follows that if the commutative group $\Gamma$ admits a height function, the torsion subgroup of $\Gamma$ must be finite.
To see this, first note that by rearranging (3) we see that $h(P) \leq h(2P)/4 + k/4$ holds for all $P \in \Gamma$. Chaining together repeated applications of this inequality (doubling $P$ each time), one deduces that $h(P) \leq \frac{h(2^m P)}{4^m} + \sum_{j=1}^m \frac{k}{4^j} = \frac{h(2^m P)}{4^m} + \frac{k}{3}(1 - \frac{1}{4^m})$ holds for all $P \in \Gamma$ and $m \geq 0$. Label the inequality just proved by (4), and let $Q \in \Gamma$ have finite order. Because the set $\{2^n Q: n \geq 1\}$ is finite, there are positive integers $r$ and $s > 1$ with $2^s 2^r Q = 2^r Q$. Applying the inequality (4) to $P = 2^r Q$ and $m = s$ and rearranging, we deduce that $h(2^r Q) \leq k/3$. Applying the inequality (4) to $P = Q$ and $m = r$, we deduce that $h(Q) \leq h(2^r Q)/4^r + (k/3)(1 - 1/4^r)$, and as $h(2^r Q) \leq k/3$ we deduce that $h(Q) \leq (k/3)/4^r + (k/3)(1 - 1/4^r) = k/3$ as desired.
We can also note that if $\Gamma$ admits a function $h$ satisfying (1), then $\Gamma = \bigcup_{n \geq 0} \{p \in \Gamma: h(p) \leq n\}$ is a countable union of finite sets and hence countable.
I do not know if every countable commutative group with finite torsion admits a height function. It seems possible that the existence of a height function also imposes other requirements (note, for example, that we have not made any use of condition (2) in the above). I would be interested in a question directed at characterizing these requirements.