Category Archives: group theory

Finding the repetend length of a decimal expansion

Posted on February 8, 2019 by Brent

We’re still trying to find the prefix length and repetend length of the decimal expansion of a fraction , that is, the length of the part before it starts repeating, and the length of the repeating part. In my previous … Continue reading →

Posted in computation, group theory, modular arithmetic, number theory, pattern | Tagged decimal, expansion, group theory, rational, repeating, repetend, totient | Comments Off

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

Origami stellated icosahedron!

Posted on February 11, 2016 by Brent

Continuing what I started in December, I finally finished making a stellated icosahedron out of 30 Sonobe units. Each Sonobe unit corresponds to an edge of the icosahedron, and interlocks with three others to give the whole thing a remarkable … Continue reading →

Posted in geometry, group theory | Tagged icosahedron, making, objects, origami, sonobe, stellated | 4 Comments

I made a Straws Thingy!

Posted on December 12, 2015 by Brent

I finished my Straws Thingy! (You can see bigger versions of all the photos here.) It’s made out of 60 straws, 12 of each color. Three straws make a triangle; four triangles of each color are woven into a tetrahedron; … Continue reading →

Posted in geometry, group theory | Tagged making, objects, straws, thingy | 4 Comments

Making stuff!

Posted on December 9, 2015 by Brent

Recently I have been on a sort of kick making physical mathematical objects. First, inspired by this post on mathcraft.wonderhowto.com, I’ve learned how to fold origami Sonobe units and assemble them into various polyhedral things. So far, I’ve made a … Continue reading →

Posted in geometry, group theory | Tagged making, objects, origami, straws | 1 Comment

MaBloWriMo 30: Cyclic subgroups

Posted on December 1, 2015 by Brent

Today, to wrap things up, we will use Lagrange’s Theorem to prove that if is an element of the group , the order of evenly divides the order of . So we have a group and an element . In … Continue reading →

Posted in algebra, group theory, proof | Tagged cyclic, groups, Lagrange, MaBloWriMo, proof, subgroups | 6 Comments

MaBloWriMo 29: Equivalence classes are cosets

Posted on November 30, 2015 by Brent

Today will conclude the proof of Lagrange’s Theorem! Recall that we defined subgroups and left cosets, and defined a certain equivalence relation on a group in terms of a subgroup . Today we’re going to show that the equivalence classes … Continue reading →

Posted in algebra, group theory, proof | Tagged classes, cosets, equivalence, groups, Lagrange, MaBloWriMo, proof | Comments Off

MaBloWriMo 28: Equivalence relations are partitions

Posted on November 29, 2015 by Brent

Today we’ll take a brief break from group theory to prove a fact about equivalence relations, namely, that they are the same as partitions. A partition is a pretty intuitive concept: you take a big set, and cut it up … Continue reading →

Posted in algebra, group theory, proof | Tagged equivalence, groups, Lagrange, MaBloWriMo, partition, proof, relation | 2 Comments

MaBloWriMo 27: From subgroups to equivalence relations

Posted on November 28, 2015 by Brent

Again, let be a group and a subgroup of . Then we can define a binary relation on elements of , called , as follows: if and only if there is some such that . That is, for any two … Continue reading →

Posted in algebra, group theory, proof | Tagged equivalence, groups, Lagrange, MaBloWriMo, proof, relation, subgroups | Comments Off

MaBloWriMo 26: Left cosets

Posted on November 27, 2015 by Brent

Let be a group and a subgroup of . Then for each element we can define a left coset of by . That is, is the set we get by combining (on the left) with every element of . For … Continue reading →

Posted in algebra, group theory, proof | Tagged cosets, groups, Lagrange, MaBloWriMo, proof, subgroups | 1 Comment