January 19th, 2013, 03:13 AM  #1 
Member Joined: Aug 2012 Posts: 72 Thanks: 0  subgroup vs. coset
What is the different between subgroup to coset?

January 19th, 2013, 10:59 AM  #2 
Senior Member Joined: Nov 2011 Posts: 100 Thanks: 0  Re: subgroup vs. coset
Here's a reference that might help http://www.math.rochester.edu/people/fa ... pak/N3.pdf If G is a group and N<G is a normal subgroup, you can consider G/N. Elements of G/N look like gN, for g in G, and are cosets. Or you can just consider cosets of a subgroup H<G as the set gH={gh\in G: h\in H}. 
January 19th, 2013, 11:13 AM  #3  
Senior Member Joined: Aug 2012 Posts: 2,306 Thanks: 706  Re: subgroup vs. coset Quote:
In the additive group of integers, the even numbers are a subgroup. The odd numbers are one of the cosets of the even numbers; but the odd numbers are not a subgroup.  
January 19th, 2013, 11:24 PM  #4 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88  Re: subgroup vs. coset
well, every subgroup is a coset (of itself) but not every coset is a subgroup. i like to think of it this way: imagine a group G is like a cake. suppose we have a subgroup H that's a certain size. if we cut G into "Hsized slices", the slices are the cosets, and the subgroup H is the slice that contains the identity. you can also think of it this way: suppose we have H living in G. the coset gH is H "translated" by g. it's easier to picture this if G is abelian, and we write g+H, instead of gH. so, for example, if G = RxR, the real plane, and * is componentwise addition, we have the subgroup: H = {(x,0) in R} (also known as "the xaxis"). in this case (a,b) + H is the set of all points (x,y) of the form (a+x,b), that is: the line y = b (which is parallel to H). it's not hard to see that: (a,b) + H = (0,b) + H, since (a,b)  (0,b) = (a,0) is in H. so (a,b) + H is just "the xaxis shifted by b". so the set of (left) cosets (RxR)/(Rx{0}) = G/H is the collection of all horizontal lines....and H is the one containing (0,0) (the one that goes through the origin). equivalently: you can view "multiplication by g" (where g is an element of G) as a FUNCTION: Lg(x) = g*x (here, Lg is supposed to remind you of L(eft multiplication by )g). so the coset gH is the image of H under Lg. for every g, the function Lg is bijective, because it has the inverse function Lg^1 (multplication by g^1): Lg(Lg^1(x)) = x = id(x). this means that gH is always "the same size" as H. now if g is in H, then g*h is in H (by closure), so for h in H, hH = H. but if g ISN'T in H, none of gH is in H. where this gets interesting is that we can have: gH = g'H, even when g does not equal g'. this happens when g = g'h (or viceversa), or put another way: g'^1g is in H. for groups: a quantity like a^1b roughly measures how "different" a and b are: if they are the same, we get e, the identity. if they are close (they lie within the same "translate" of H), then a^1b is in H (which you can think of as being "the closest coset to home", where "home" here means the identity of the group). 

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