Theorem 26: the first isomorphism theorem

Sorry for the gap; things have been pretty busy here.  The good news, though, is that it’s time for another theorem!

I’ve written previously about groups, when I wrote about Lagrange’s theorem in group theory.  Today I’d like to talk some more about groups, so if you don’t know what one is I suggest you pause for a moment and go back to that post.

I’d like to start with a particular example.  Let’s think about two groups.  One is , the group of integers under addition.  The other is the group of integers under addition (mod 7), which I’ll write as .  How are they related?  They’re very similar; the only real difference is that in , we treat two numbers as the same if they differ by a multiple of 7, because we’re adding (mod 7).  In fact, that’s a precise recipe for how to get from , isn’t it?

This leads us to the idea of a quotient group.   Here, we take the multiples of 7 (which we could write as ), and think of them as being all the same (and in particular the same as 0).  Then we say that two elements of are the same if they differ by a multiple of 7.  (E.g. we think of 6 and 13 as being the same.)  This gives us 7 classes of elements (all the elements in a single class are the same in this sense).  We pick a representative of each class (conventionally 0, 1, 2, 3, 4, 5, 6).  And it turns out that when we do this, with the same group operation as before (addition), we do get something that’s a group.  We write it as the quotient .

What happens if we’re a bit more general?  Let’s take a group , and a subgroup .   What does it mean to quotient by , and is the resulting object always a group?

Theorem 10: Lagrange’s theorem in group theory

The term has just started, and so I have been contemplating the exciting mathematics in store for the new first years.  This week I thought I’d tell you about one such result.

New students often find that some of the mathematics they meet at university is more abstract than the mathematics they studied at school.  Abstraction can be an extremely useful tool.  Broadly speaking, here’s what happens.  Mathematicians notice several examples of the same behaviour.  They want to explore why: what is it about those situations that leads to the same behaviour?  They try to write down a list of those key properties.  They then define a new object: it’s anything that has those properties.  (So all of the initial situations should be examples of this new kind of object.)  That should hopefully lead to a better understanding of what’s going on.  Moreover, if mathematicians can prove something about this object using only the knowledge that the object has those key properties, then they have proved a result about every example of that object — all in one go!

One good example of this is the notion of a group.  I’m not going to go through the definition of a group in great detail here, so if you haven’t come across the concept before then you might like to read this article on NRICH.  Here, I’ll remind you of the definition of a group and give a few examples.