# Gracious Living

Symmetric Groups
December 18, 2010, 11:36
Filed under: Algebra, Math, Uncategorized | Tags: , , , ,

We’ve seen symmetric groups before.  The symmetric group on an arbitrary set, $S_X$ or ${\rm Sym}(X)$, is the group of bijections from the set to itself.  As usual, we’re only interested in the finite case $S_n$, which we call the symmetric group on $n$ symbols.  These are pretty important finite groups, and so I hope you’ll accept my apology for writing a post just about their internal structure.  The language we use to talk about symmetric groups ends up popping up all the time.

Orbits, Stabilizers, and Conjugacy Classes
December 1, 2010, 23:45
Filed under: Algebra, Math | Tags: , , ,

We’ve seen a couple of ways to cut a group into pieces. First, we can look at its subgroups, which I visualize as irregular blobs all containing the identity. Under inclusion, these subgroups form a lattice, a partially ordered set in which every two elements have a greatest lower bound (here their intersection) and a least upper bound (here the group generated by their union). The structure of this lattice reveals a lot about the structure of the group and the things attached to it, the fundamental theorem of Galois theory being one powerful example. Second, given one subgroup, we can look at its cosets, which I visualize as parallel slices, and the quotient groups they form.

But cosets are tied to a specific subgroup and aren’t groups themselves, and the lattice of subgroups is in a sense too much information. One of the common problems of math is to find invariants — simpler objects that encode a lot of the data in a given structure and are easier to find. The only real way to get simpler than a group is with numbers, and one sequence of numbers is the class equation, which describes the conjugacy classes of the group. I visualize these as radial slices, like the layers of an onion.