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Category Archives: group theory
Finding the repetend length of a decimal expansion
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 on Finding the repetend length of a decimal expansion
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
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Origami stellated icosahedron!
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!
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
Making stuff!
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
MaBloWriMo 30: Cyclic subgroups
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
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
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MaBloWriMo 28: Equivalence relations are partitions
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
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 on MaBloWriMo 27: From subgroups to equivalence relations
MaBloWriMo 26: Left cosets
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