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 formed by the elements $g$ such that $2g = 0$. a) Now prove that $x = \sum_{g \in G_2} g$. b) Now prove that $x = 0$ if Card($G_2$) $\neq 2$. If Card($G_2$) = 2, now prove that $x$ is the unique nonzero element of $G_2$. c) Now prove that Card($G_2$) = 2 if and only if the 2-Sylow subgroup of G is cyclic and $\neq 0$. d) \* Let $p$ be a prime number. Now prove that

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