Little post.  Because I love doing things that comments tell me to do, we’re going to use Lagrange to prove a neato theorem.  Now, normally, if I told you, “Hey, guy, I’ve got a group G with n elements.  What one is it?” you’d probably be unable to tell me!  Why?  Lots of different groups have the same order!  For example, if we’re talking about order 8, are we talkin’ D_{8}?  Are we talkin’ Z_{8}?  Are we talkin’ Q_{8}?  I just don’t know!

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How could I have been so naive?  How could I have been so myopic?  How is it that I thought I could just wrap up group theory without mentioning Lagrange’s theorem?  How could I let this topic die out not with a bang but with a whimper?

Let us, for old time’s sake, state one more theorem for the group theory primer — and this one’s a biggie!  Remember how division is defined for rational numbers?  \frac{a}{b} sort of means “split a into little piles of size b, and \frac{a}{b} is how many piles there are.”  For example, if we have 12 batteries and put them into piles of 3 batteries each, how many piles do we have?  This doesn’t take a rocket scientist.

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Last time we talked about a whole lot of stuff.  We did homomorphisms, isomorphisms, and talked about the first ismorphism theorem.  What did this one state?  It states that for G,H are groups and f:G\rightarrow H is a homomorphism, then we have that G/Ker(f) \cong Im(f), or, in other words, the quotient of G with the kernel of the map is equal to the image of the map.  This makes sense if you think about it: we’re kind of condensing everything that goes to 0 when we map it away from G and we say that these elements ultimately don’t matter in the image — but, because of the nice properties of homomorphisms, a lot of other elements map onto each other, too.

Today, we’re going to discuss the final two isomorphism theorems (which don’t come up as often, but they’re nice) and conclude with one of the most used theorems in elementary abstract algebra: Cauchy’s Theorem.

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Last time we went over some normal subgroups, how to direct product two groups, and how to quotient out by (normal) subgroups.  As we said before, though, groups (like vector spaces) are pretty boring by themselves.  Yes, studying groups by themselves can give us relations between elements and so on (like what kinds of elements in a particular group have the property such that when you square them they become the identity), but, like vector spaces, we can learn a lot about a group by what it can and can’t map into nicely.

Now, let’s think about this for a second.  What if I said something like the following: let’s take a group G such that the elements are \mbox{possum} and \mbox{fox}.  Let’s say that

\mbox{possum}\times \mbox{possum} = \mbox{possum}

\mbox{possum} \times \mbox {fox} = \mbox{fox} \times \mbox{possum} = \mbox{possum}

\mbox{fox} \times \mbox{fox} = \mbox{fox}

and those are all the possible interactions.  You could give any reasonable justification to this group, but it reduces to the fact that it is just a group: it’s just a set of elements and an operation.

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If we think about groups as if they were numbers, we’d want to add, subtract, multiply, and divide stuff.  Unfortunately, groups aren’t as simple as numbers, and we have more complex notions of what all of these things should correspond to.

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Last time, we talked about what a group is.  This time, we’ll go over some specific groups.  In the next post, we’re going to go over some basic theorems about groups.

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Personal Motivation: This morning I awoke from a dream and all I could think about was manipulating group elements as if they were linear maps.  We’ve been talking about linear maps a lot, and one of their nice properties is that they can be represented by a matrix; if we were to represent group elements as matrices, then we would be able to use a lot of the linear algebra we know to prove a few things about groups!  In fact, this type of thinking has a name: representation theory.  I won’t lie to you, readers: I’ve taken a class in this, but I hated it and paid very little attention in it.  Despite this, I’m going to begin going over the text and select some nice theorems to write about.

What I’m actually going to write about in this post: Because groups are so damn important in abstract algebra, I’m going to take this post to construct them.  Because this would be quite boring to the general mathematician who has already taken abstract algebra, I’m going to do it in a slightly weird way: I’m going to build them up by adding structure to sets.

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We did rank-nullity last time, and this time we’re going to go over a few cool applications to rank-nullity.

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The last legit theorem we went over was the rank-nullity theorem.  This time, we’re going to discuss a couple of cute theorems dealing with the rank-nullity theorem and its applications to vector spaces.  For this post’s sake, all vector spaces will be finite.

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This post is going to be a short post, and a deviation from what I’ve normally written about; I have been taking a slight break from linear algebra, but expect a post soon regarding vector spaces and lots of stuff we can do with them.

The title of this post comes from a student that I had recently, asking why “squaring both sides” of an equations is legitimate.

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