I am trying to prove Cauchy theorem for groups which states that:
If $G$ is a finite group, $|G| =n$, and $p$ a prime number which is a divisor of $n$, then there exists $g\in G$ such that $g^p =1$.
I am following a proof which defines an action over the following set:
$X = \{ (a_1,\cdots,a_p): a_1\cdots a_p =1, a_i\in G, i =1,\cdots,p\}$.
Then we define an action of $(\mathbb{Z_p},+)$ over $X$ by: $t\in \mathbb{Z_p},$
$t\cdot(a_1,\cdots,a_p) = (a_{t+1},a_{t+2},\cdots,a_t) $ where the subindexes in the second equality are taken modulo $p$.
I can see why the action shall be well-defined: for example, since $a_1\cdots a_p = 1$, this implies that $a_2\cdots a_{p} = a_{1}^{-1}$, which also implies that $a_2\cdots a_pa_{1} = 1$, that is, $1\cdot(a_1,\cdots,a_p)\in X.$
But how to prove it with more generality?
Thanks in advance!