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# Tag Archives: order

## Fermat’s Little Theorem: proof by group theory

It’s time for our third and final proof of Fermat’s Little Theorem, this time using some group theory. This proof is probably the shortest—explaining this proof to a professional mathematician would probably take only a single sentence—but requires you to … Continue reading

Posted in group theory, number theory, primes, proof
Tagged combinatorics, group, order, proof, theory

## MaBloWriMo 23: contradiction!

So, where are we? We assumed that is divisible by , but is not prime. We picked a divisor of and used it to define a group , and yesterday we showed that has order in . Today we’ll use … Continue reading

Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged contradiction, groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
5 Comments

## MaBloWriMo 22: the order of omega, part II

Yesterday, from the assumption that is divisible by , we deduced the equations and which hold in the group . So what do these tell us about the order of ? Well, first of all, the second equation tells us … Continue reading

Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
1 Comment

## MaBloWriMo 21: the order of omega, part I

Now we’re going to figure out the order of in the group . Remember that we started by assuming that passed the Lucas-Lehmer test, that is, that is divisible by . Remember that we also showed for all . In … Continue reading

Posted in algebra, group theory, modular arithmetic, number theory, proof
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, order, prime, proof, test, X
2 Comments

## MaBloWriMo 15: One more fact about element orders

I almost forgot, but there is one more fact about the order of elements in a group that we will need. Suppose we have some and we happen to know that is the identity. What can we say about the … Continue reading

Posted in algebra, group theory, number theory, proof
Tagged elements, finite, groups, MaBloWriMo, order
1 Comment

## MaBloWriMo 14: Element orders are no greater than group size

Today we will give an answer to the question: What is the relationship between the order of a group and the orders of its elements? Yesterday, I claimed we would prove that for any element of a group , it … Continue reading

Posted in algebra, group theory, proof
Tagged elements, finite, groups, MaBloWriMo, order
2 Comments

## MaBloWriMo 13: Elements of finite groups have an order

Recall from yesterday that if is a group and is some element of the group, the order of is defined as the smallest number of copies of which combine to yield the identity element. I forgot to mention it yesterday, … Continue reading

Posted in algebra, group theory, proof
Tagged elements, finite, groups, MaBloWriMo, order