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January 19th, 2013, 03:13 AM   #1
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subgroup vs. coset

What is the different between subgroup to coset?
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January 19th, 2013, 10:59 AM   #2
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Re: subgroup vs. coset

Here's a reference that might help ... 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}.
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January 19th, 2013, 11:13 AM   #3
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Re: subgroup vs. coset

Originally Posted by goodfeeling
What is the different between subgroup to coset?
Perhaps an example will help.

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.
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January 19th, 2013, 11:24 PM   #4
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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 "H-sized 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 component-wise addition, we have the subgroup:

H = {(x,0) in R} (also known as "the x-axis").

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 x-axis 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 vice-versa), 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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