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Tag Archives: order
Order of operations considered harmful
[The title is a halfjoking reference to Edsger Dijkstra’s classic paper, Go To Statement Considered Harmful; see here for more context.] Everyone is probably familiar with the socalled “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
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 on Fermat’s Little Theorem: proof by group 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 LucasLehmer 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
Comments Off on MaBloWriMo 13: Elements of finite groups have an order
MaBloWriMo 12: Groups and Order
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
A computerchecked proof of the odd order theorem
Big news: a proof of the FeitThompson 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