Question: Let $G$ be a group. Suppose $x$ has finite order $n$. Show that $x^k=1$ if and only if $k$ is a multiple of $n$.

Intuitively, I think I understand this, but I am not sure how to prove it. Any help is appreciated.

Answer: 
In math, the fact that a statement $A$ implies a statement $B$ is written this way: $A \implies B$

The meaning of $A \implies B$ is defined by this truth table: 

$$
 \begin{matrix}
 A & B & | & A \implies B\\
 T & T & | &T\\ 
 T & F & | &F\\
 F & T & | &T\\
 F & F & | &T\\
 \end{matrix}
$$

That means that $A \implies $B is false only when $A$ is true and $B$ is false. In the other cases $A \implies $B is true.

So let's say I have a theorem that states "If $n$ is a multiple of $6$ then it must be a multiple of $2$" ($n$ multiple of $6 \implies n$ multiple of $2$). For this theorem to be true, if you encounter a multiple of $6$ then it must be also a multiple of $2$ because if not the implication would be false. But if you encounter a number that is not a multiple of $6$ then it can be a multiple of $2$ or not without invalidating the theorem, because the theorem says nothing about numbers that are not multiple of $6$.

You can also see what the implication means by looking at sets: 
[![enter image description here](https://i.stack.imgur.com/U9naZ.png)](https://i.stack.imgur.com/U9naZ.png)
Fundamentally, mathematics is a guide to reducing the unknown. When we have a statement "If $P$ then $Q$", that means if we somehow already know that $P$ is true, then we now can be assured that $Q$ is true. Each theorem that proves an implication allows us to expand our knowledge of true facts by using chains of implications. We start with known supposed true facts and, using chains of implications, we can conclude many other true facts on the basis of the suppositions. And finally arriving at what we want to prove to be true also. The Achilles' heel of this method is that we are dependant of the starting true facts, and if any one of them turns out to be false, then the results of our implications are now not assured to be true. Another weakness is that we could make several kinds of errors in applying the implications because "to err is human" and "a chain is as strong as its weakest link".