Normal Subgroups and Factor Groups


You first encounter factoring in arithmetic. For example, 48 is 6x8. (It's also 3x16 and 2x24, but let's not get distracted.) This idea of breaking an object (integers) into smaller objects (smaller integers) by an operation (multiplication) is a powerful way to dissect objects to find the key building blocks. But is there something like factoring for groups? There is!


Suppose a group G has a proper subgroup H. The mathematical way to write this is as H < G. (One example to keep in mind would be the group of integers Z under addition with the subgroup of multiples of 10.) You can use H to partition G into non-overlapping sets called cosets. To do this, just pick an element g1 in the group G that's not in H, then look at the set g1 + H. You can then keep going by choosing an element g2 that's not in the subgroup H or the set g1 + H. Using this, you get a third set g2 + H. It takes a little work to see that these sets do not overlap, but it's worth playing around to see why this is the case. But back to the definition! Remember I asked you to keep in mind the example of integers and subgroup of multiples of 10? In that example, the subgroup H is the multiples of 10, and the cosets are 1 + 10Z, 2 + 10Z, ..., 9 + 10Z. These are the integers with remainders of 1, 2, ..., 9 respectively. It turns out that you can treat these cosets as elements in a new group - the factor group. For the example of G = Z and H = 10Z, the cosets are the congruence classes mod 10.

The idea of treating cosets as elements in a new factor group is a big leap in abstraction. You could say that Gauss pioneered this idea when he introduced modular arithmetic. However, he was not thinking this abstractly at the time. But modular arithmetic proved to be so successful it was only natural to see how well this trick could be used in other places. But, back to the definition!

The coset trick doesn't always work. In fact, it only works for specific subgroups H. When this trick does work, we say H is a normal subgroup of G. (When this trick doesn't work, we simply call H a subgroup of G...) If H is a normal subgroup of G, then we can treat the cosets as elements in a new group - the factor group. The math notation for the factor group is G/H. This notation should remind you of division! This is because you're using H to "divide" G into cosets.


We actually just saw an example! G = Z under addition and H = 10Z. This gives you the integers mod 10...