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# Category Archives: group theory

## MaBloWriMo 20: the group X star

So, where are we? Recall that we are assuming (in order to get a contradiction) that is not prime, and we picked a smallish divisor (“smallish” meaning ). We then defined the set as that is, combinations of and where … Continue reading

Posted in algebra, arithmetic, group theory, number theory
Tagged groups, MaBloWriMo, monoids, X

## MaBloWriMo 19: groups from monoids

So, you have a monoid, that is, a set with an associative binary operation that has an identity element. But not all elements have inverses, so it is not a group. Assuming you really want a group, what can you … Continue reading

Posted in algebra, group theory, proof
Tagged elements, groups, inverses, MaBloWriMo, monoids, proof
1 Comment

## MaBloWriMo 18: X is not a group

Yesterday we defined along with a binary operation which works by multiplying and reducing coefficients . So, is this a group? Well, let’s check: It’s a bit tedious to prove formally, but the binary operation is in fact associative. Intuitively … Continue reading

Posted in algebra, arithmetic, group theory, number theory
Tagged groups, MaBloWriMo, monoids, X
1 Comment

## MaBloWriMo 17: X marks the spot

Recall that we are trying to prove that if is divisible by , then is prime. So let’s suppose is divisible by . We’ll prove this by contradiction, so suppose is not prime: if we can derive a contradiction, then … Continue reading

Posted in algebra, arithmetic, group theory, number theory
Tagged groups, MaBloWriMo, omega, proof, X
3 Comments

## MaBloWriMo 16: Recap and outline

We have now established all the facts we will need about groups, and have incidentally just passed the halfway point of MaBloWriMo. This feels like a good time to take a step back and outline what we’ve done so far … Continue reading

Posted in algebra, arithmetic, computation, famous numbers, group theory, iteration, modular arithmetic, number theory, primes
Tagged groups, lehmer, lucas, MaBloWriMo, Mersenne, omega, prime, proof, summary, test
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