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Monthly Archives: November 2015
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
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
MaBloWriMo 11: Examples of Groups
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
Posted in algebra, group theory, modular arithmetic, number theory
Tagged examples, groups, MaBloWriMo
5 Comments
MaBloWriMo 10: Groups
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
Posted in algebra, group theory
Tagged associativity, combining, definition, groups, inverses, MaBloWriMo
2 Comments