I am reading Mordell's theorem from Silverman and Tate's book. It uses the notation "$2 \Gamma$" for any commutative group $\Gamma$. Does it mean the group $\{g+g\mid g\in \Gamma \}$?
Theorem 3.5 (Descent Theorem). Let $\Gamma$ be a commutative group and suppose that there is a function $h : \Gamma \to [0, \infty)$ with the following three properties:
(a) For every real number M, the set $\{P \in \Gamma : h(P) \leq M\}$ is finite.
(b) For every $P_0 \in \Gamma$ there is a constant $\kappa_0$ so that $h(P+P_0) \leq 2h(P) + \kappa_0$ for all $P \in \Gamma$.
(c) There is a constant $\kappa$ so that $h(2P) \geq 4h(P)- \kappa $ for all $P \in \Gamma$.
Suppose further that
(d) The subgroup $2\Gamma$ has finite index in $\Gamma$.
Then, $\Gamma$ is finitely generated.