Exercise 20, § 6

Let $G$ be a finite commutative group written additively. Let $x = \sum_{g \in G} g$ and let $G_2$ be the subgroup of $G$ consisting of the elements $g$ such that $2g = 0$. (a) Show that $x = \sum_{g \in G_2} g$. (b) Show that $x = 0$ if $\mathrm{Card}(G_2) \neq 2$. If $\mathrm{Card}(G_2) = 2$, show that $x$ is the unique non-zero element of $G_2$. (c) Show that $\mathrm{Card}(G_2) = 2$ if and only if the Sylow 2-subgroup of $G$ is cyclic and $\neq 0$. (d) *Let $p$ be a prime number. Show that

(Apply (b) to the multiplicative group of the field $\mathbf{Z}/p\mathbf{Z}$.)