Show that any commutative group of finite order is finitely generated $\mathbb{Z} $-module .

Question

Show that any commutative group of finite order is finitely generated $\mathbb{Z} $-module .

I see that any commutative group is $\mathbb{Z} $-module. But why is any finite group finitely generated?

Answer 1

But why is any finite group finitely generated?

Let $(G, \circ)$ be a finite group. Then $G=\{g_1, g_2, \dots, g_{\lvert G\rvert}\}$ for some $g_i$ because $G$ is finite, where $\lvert G\rvert$ is the order of the group. Every element of $(G, \circ)$ can be written as a product of elements in $G$ with respect to $\circ$; namely, $g_i=g_i$ for all subscripts $i$. Hence $(G,\circ)$ is finitely generated.