Here's some intuition on the meaning of a factor group.
Most people learning abstract algebra, as far as I can tell, have no idea why homomorphisms and factor groups are sensible things to think about. They quickly come to understand the idea of a group, and enough varied examples are usually given that they can see how the idea of a group applies to a number of things. They quickly come to terms with subgroups, though the idea looks rather trivial to them. Then you get to homomorphisms and factor groups; at this point, most classes run out of intuition and just jump in for some unmotivated mathematical constructions.
I’m not quite sure why this is, honestly. Anyone with the slightest modicum of mathematical curiosity probably has thoughtn about factor groups since they were seven or eight years old. In the context of integers and addition, most children realize on their own (whether it’s taught to them or not) that the sum of two even numbers, or of two odd numbers, is even, while the sum of an even number and an odd number is odd. This is, of course, a factor group. Students who are presented with the mathematical definition of a factor group should first have, in their set of mental tools, this simple intuitive definition:
Factor Group: For any group (G,*), a factor group is a group that is obtained by being sufficiently sleep-deprived (or perhaps drunk, depending on the university) that one can’t tell the difference between some members of the original group, and then trying to write down a group table.
Of course, one then goes on to point out that sometimes this works, but sometimes it doesn’t. If one looks at the integers and only sees “even” or “odd”, then it works. If one looks at the integers and only sees “negative” or “non-negative”, then it doesn’t work, since the sum of a negative number and a positive number could be either negative or positive. It then becomes natural to ask when it works, and when it doesn’t. This provides a justification, then, for nailing down the abstract definitions, defining normal subgroups, proving that the factor group is well-defined when modding out a normal subgroup, and so on. First, though, the student needs to be convinced that these are natural things to think about.
Speaking of defining normal subgroups, it is really inexcusable how many students have never even noticed the close relationship between normal subgroups and commutativity. Sure, everyone knows that all subgroups of an abelian group are normal; but this seems to be treated as a sort of occasionally useful curiosity. Few students are even exposed to the simple fact that normality of subgroups is inextricably entwined in the degree to which the subgroup commutes with the surrounding group.
A quotient group is a shining
example of the beauty of abstract algebra. We’ve abstracted away from numbers to these
crazy group things, and one reward is that we can see what division really means. It’s
more than just a simple bit of arithmetic: division is a way of describing a fundamental
structural relationship that pervades mathematics.
So what is division all about?
Suppose you want to divide 50 by 5. What you’re really doing is saying
you’ve got a collection of 50 indistinguishable things, and you want to break it into a 5
indistinguishable collections. Since you started with 50 indistinguishable things, that means
that you’ll end up with 5 sets of 10 things.
That’s pretty simple, right? Now, suppose that we’re not talking about simple numbers. Instead we
want to work in terms of groups. Can we take that basic concept of division, and find some meaningful way of applying it to groups?
Well, first, we need to somehow talk about division in a way that doesn’t involve numbers. We start with a group – that is, a collection of objects with some kind of meaningful structure. What can we divide it by? A group with a similar structure – in fact, a group with the same structure: a subgroup But not just any subgroup: it’s got to be a normal subgroup, because that’s the kind of subgroup that properly preserves the structure of the group.
So what happens when we divide a group by one of its normal subgroups? We partition the
group into a new group, where the elements of the new group are formed from subsets of the
elements of the original group. It’s the same idea as simple integer division described up above, except that we want to preserve the group structure, so the result is going to be a group.