I was provided this difficulty for homework and I to be not certain where to start. I know a systems using Lagrange"s theorem, yet we have actually not proven Lagrange"s theorem yet, in reality our teacher hasn"t also mentioned it, so i am guessing there should be another solution. The just thing I could think that was showing that a team of element order $p$ is isomorphic to $ asiilaq.netbbZ/p asiilaq.netbbZ$. Would this work?

Any guidance would be appreciated.

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As video camer McLeman comments, Lagranges theorem is substantially simpler for groups of element order than for general groups: it states that the team (of prime order) has no non-trivial ideal subgroups.

I"ll usage the complying with

## Lemma

Let $G$ be a group, $xin G$, $a,bin asiilaq.netbb Z$ and $aperp b$. If $x^a = x^b$, then $x=1$.

*Proof*: by Bezout"s lemma, part $k,ellin asiilaq.netbb Z$ exist, such that $ak+bell=1$. Then $$ x = x^ak+bell = (x^a)^k cdot (x^b)^ell = 1^k cdot 1^ell = 1 $$

(If you know a tiny ring theory, you can prefer to an alert that the set $ x^i=1\subseteq asiilaq.netbb Z$ forms perfect which need to contain $(a,b)=1$ if it consists of $a$ and $b$.)

## The question

Now permit $P$ it is in an arbitrary group of element order $p$. Consider any type of $xin P$ such the $x
eq 1$ and consider the set$$ S = 1, x, x^2 , dots , x^p-1 \subseteq P.$$First assume two of these aspects are equal, say $x^u=x^v$ and $u

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