When the subgroup H is normal, the cosets form their own group. The product of two cosets is found by multiplying any two cosreps together, and taking that coset. We need to prove this is well defined, i.e. you always get the same coset no matter which cosreps you select. If, instead of xy, we had selected xuyv, where u and v come from H, we replace uy with yw, since the left and right cosets of y are the same. Now wv lives in H, so we have found another member in the same coset, the coset of xy. We needed H to be normal, to replace uy with yw.
The identity represents the identity coset, and a coset's inverse is found by inverting one of its cosreps. The resulting group of cosets is called the quotient group, or the factor group, and H is called the kernel. In fact, we will now switch notation, whence K is the kernel and H is the quotient group. Hope this isn't too confusing.
|
You'll notice that the quotient retains the coarse structure of the group,
without all the fine detail;
like looking at an object through the wrong end of the telescope.
Z15 has a normal subgroup of 0,5,10,
and a quotient group with cosreps 0,1,2,3,4, i.e. Z5.
The quotient tells us that the original group has some similarity to the integers mod 5;
the kernel tells us the group has something to do with Z3.
Of course we could have used 0,3,6,9,12 as kernel,
making Z3 the quotient.
We don't always have the option to switch kernel and quotient,
but in this case we do.
If G consists of the powers of a generator x, G is cyclic. The only cyclic groups are the integers and the integers mod n, for all positive values of n. The finite cyclic groups are denoted Zn. These groups are simple iff n is prime. If n = pq, G can have the factor group Zp or Zq, as demonstrated by Z15 above. |
|