Stuck in proving the Fermat's little theorem (with group theory)

Question

I am having trouble proving Fermat's little theorem using group theory. These are my steps so far:

Let $(Z_p,+,\times)$ a Field, and we assume we know that the non-zero elements form a Group multiplicatively of order $p-1$, and we know that the order for all $a$ in the group $(Z_p,\times)$, is $x$ such that $a^x\equiv 1\pmod p$. We need to show that $x=p-1$. Well, the subgroup generated by $a$ of order $x$ must divide the order of the group $p-1$, so $x|p-1$ by Lagrange, and so $p-1=x*m$ for some $m$ positive integer. So $a^{p-1}=a^{x*m}=1^m=1\pmod p$.

This proof feels sloppy and even wrong, but it's the best way I can word out my intuition, may you please help me out?

Answer 1

This is basically correct, but some of your wording is confusing and can be cleaned up. Specifically, the following part:

we know that the order for all $a$ in the group $(Z_p,\times)$, is $x$ such that $a^x\equiv 1\pmod p$. We need to show that $x=p-1$.

Here you are being very ambiguous about what $x$ represents or what you want to be true of $x$. Is $x$ supposed to be the order of every element, or just of one particular element $a$? Also, if $x$ is the order of $a$, then $a^x\equiv 1\pmod p$, but the converse is not true. So the property that $a^x\equiv 1\pmod p$ does not uniquely define the order as you seem to be implying in the first sentence. On the other hand, if you are defining $x$ as the order, then the second sentence is just wrong: you don't want to prove that $x=p-1$. Instead, you want to show that $a^{p-1}\equiv 1\pmod p$, which does not necessarily mean that $x=p-1$.

I might rephrase this part as follows:

Let $a\in Z_p$ and let $x$ be the order of $a$. Then $a^x\equiv 1\pmod p$. We need to show that $a^{p-1}\equiv 1\pmod p$.

In particular, notice how my first sentence clearly and unambiguously introduces a specific element $a$ and defines $x$ as the order of $a$, rather than presenting a jumble of facts about $x$ indirectly. Only once the definition is made do I start discussing other facts.