Monthly Archives: November 2015

MaBloWriMo 19: groups from monoids

Posted on November 20, 2015 by Brent

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 →

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MaBloWriMo 18: X is not a group

Posted on November 19, 2015 by Brent

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 →

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MaBloWriMo 17: X marks the spot

Posted on November 18, 2015 by Brent

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 →

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MaBloWriMo 16: Recap and outline

Posted on November 17, 2015 by Brent

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 , , , , , , , , , | 2 Comments

MaBloWriMo 15: One more fact about element orders

Posted on November 16, 2015 by Brent

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 →

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MaBloWriMo 14: Element orders are no greater than group size

Posted on November 15, 2015 by Brent

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 →

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MaBloWriMo 13: Elements of finite groups have an order

Posted on November 14, 2015 by Brent

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 →

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MaBloWriMo 12: Groups and Order

Posted on November 13, 2015 by Brent

Continuing our discussion of groups (see here and here), today I want to discuss the concept of order, which is defined both for groups themselves and for the elements of a group. The order of a group simply means the … Continue reading →

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MaBloWriMo 11: Examples of Groups

Posted on November 12, 2015 by Brent

For reference, here’s the definition of a group again: a set a special element a binary operation on such that is associative, that is, whenever , , and are elements of is the identity for , that is, for every … Continue reading →

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MaBloWriMo 10: Groups

Posted on November 11, 2015 by Brent

So what is a group? Intuitively, a group consists of a set of things , such that there is a way to combine any two things together there is a special thing which has no effect when combined with other … Continue reading →

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