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April 20th, 2010, 12:26 PM   #1
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Finding the Group G/H (H being a subgroup of G)

Does anybody know a general method to find the Group G/H (Where G is a Group and H is a subgroup of G)

For example

(1) What is the group S3/H ?????

S3 = {e, a, a^2, b, ab, (a^2)b} (Permutation group of order 6)
H =< a >= {e, a, a^2} is a cyclic subgroup of G

(2) What is the group GL(n,R)/GL+(n;R)???

GL(n,R) = { nxn Matrices with real entries whose determinants are not zero}
GL+(n;R)= { nxn matrices with real entries whose determiants are positive}

(3) Consider the dihedral group D4 =< a, b >= {e, a, a^2, a^3, b, ab, (a^2)b, (a^3)b}
Find the groups D4/H where H is a normal proper subgroup of D4

(4) Consider S to be the set of all transformations on R such that if x belongs to R
s : x --> x' = ax + b
with a, b real numbers and a not = 0.
Let S1 be all transformations of the form x --> x' = x+b and S2 be all transformations
of the form x --> x' = ax
S/S1 is a group, which group is it?

Help will be greatly appreciated
millwallcrazy is offline  
 
April 26th, 2010, 07:18 AM   #2
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Re: Finding the Group G/H (H being a subgroup of G)

G/H refers to (generally) the set of left cosets of H in G. In your first question, S3/<a>, we know several things. We know that [G:<a>] = 2. So we know that G/H has order 2. Of course, there is really only one order 2 group, which I will call Z2 (also frequently called C2). What is it?

Well, if you really have no idea, start multiplying H by elements of G. The claim is that in those 6 products, 3 will equal something, and 3 will equal something else (this is also the spirit of Lagrange's Theorem, as the size of cosets are equal). Now, knowing the answer, I know this conjecture is true. Although I also know that S3 is a semidirect product of A3 and Z2, so even going in I have an advantage. But if you are stuck, multiply them out.

The second question is more interesting than the first, but the same basic idea will work. It is important to now that you will never get 0 anywhere. If you are particularly stuck, then I will bring attention to something frequently called the 'sign homomorphism' - which is far beyond the idea of cosets and quotient groups but is instructive nonetheless. If you consider the map F from GL --> {1, -1}, where F(A) = det(A) for all A in GL, then we note that the set of things mapped to {1} are those matrices with positive determinants. It also happens that this set, GL+ is a subgroup of GL (as is always the case in such maps, here GL+ is called the kernel of this transformation). Either this was very helpful or very useless - if the latter, don't worry about it and try other methods of reasoning.

The third question is cute. I note that you should know that any subgroup N of a group G such that [G:N] = 2 is normal in G. With this, you should consider the subgroups. How many are there? And how many 'types' of subgroup are there?

The fourth you could perhaps guess. Now this does not mean that you shouldn't explain it, but it is somehow not as difficult as it might appear. First, do the other three, and note that the methods of the first two are essentially the same as the methods here.

Good luck! Helpful?
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