Tag Archives: order

Order of operations considered harmful

Posted on October 7, 2019 by Brent

[The title is a half-joking reference to Edsger Dijkstra’s classic paper, Go To Statement Considered Harmful; see here for more context.] Everyone is probably familiar with the so-called “order of operations”, which is a collection of rules that reflect conventions … Continue reading →

Posted in arithmetic, teaching | Tagged harmful, operations, order, parentheses, parsing, perform, trees | 9 Comments

Fermat’s Little Theorem: proof by group theory

Posted on December 21, 2017 by Brent

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 | Comments Off

MaBloWriMo 23: contradiction!

Posted on November 24, 2015 by Brent

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

Posted on November 23, 2015 by Brent

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

Posted on November 22, 2015 by Brent

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

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 →

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

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 →

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

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 →

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

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 →

Posted in algebra, group theory | Tagged groups, MaBloWriMo, order | 1 Comment

A computer-checked proof of the odd order theorem

Posted on November 11, 2012 by Brent

Big news: a proof of the Feit-Thompson Theorem (also known as the “odd order theorem”) has been completely formalized and verified by a computer, using the Coq proof assistant! Wait, what? Huh? you’re probably thinking. Well, let me unpack that … Continue reading →

Posted in algebra, links, programming, proof | Tagged assistant, Coq, Feit, odd, order, proof, Thompson | 10 Comments