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# Order of a group’s element

Let be a group and such that and

(1)

where is the identity element of the group. Find the order of .

Solution

We will begin stating a lemma:

Lemma: If then .

Proof:

First we multiply with and from right and left respectively. Thus one can see that

(2)

Thus

(3)

and

(4)

Now, we use the main relation and so

By repeating the previous procedure, one can prove the result.

Using the lemma we see that and thus . Since is prime the order of will be either or .

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