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It should be common knowledge now that I have trouble with Group Theory. I would like to go back and start from the beginning but I haven't the luxury of time at this point. So for the present time I am resigned to just keeping up with the class the best I can. For anyone has the time and patience, I would appreciate it if someone can look over my work for the following 2 questions. Advice on how to approach the question, hints, interpretations of concepts, and expansions on concepts are welcome.

Question1:

Let G = D_{6}= {u, y, y^{2}, x, xy, xy^{2}} where x^{2}= u, y^{3}= u, and yx = xy^{-1}. Let H = {u,x}. (u = the identity element).

i) Write down the elements of the right cosets A = Hy and B = Hy^{2}.

ii) Calculate the product AB = (Hy)(Hy^{2}) of the cosets Hy and Hy^{2}(ie., write down and simplify every possible product ab, where a is an element of A and b is an element of B).

iii) Is AB a coset of H in G?

iv) Is AB a coset of any subgroup of G? (Hint: Use Lagrange's Theorem).

i)

A = Hy = {uy, xy} = {y, xy}

B = Hy^{2}= {uy^{2}, xy^{2}} = {y^{2}, xy^{2}}.

ii)

y*y^{2}= y^{3}= u

y*xy^{2}= xy^{2}

xy*y^{2}= xy^{3}= xu = x

xy*xy^{2}= x*xy^{-1}*y^{2}= u*y = y

Therefore AB = {u, y, x, xy^{2}}

iii)

I am not sure about this part of question 1, but I would think that AB is not a coset of H in G since AB has 4 elements while H has only two.

iv)

I am also not sure about this part of question 1. However, I think that since the order of AB is 4 and that the order of G is 6, 4 is not a divisor of 6 hence AB cannot be a subgroup of G. If there cannot be a subgroup of order 4, AB cannot be a coset of any subgroup since there are no subgroup of order 4. (?)

Question2:

i) Let G be a group, and let H be a subgroup of G. What condition tells you that H is a normal subgroup of G?

ii) Prove the following: H is normal in G iff g^{-1}Hg = H for every g which is an element of G.

i)

A subgroup H of a group G is a normal subgroup of G if the following is true:

Condition: gH = Hg for every g which is an element of G. That is, the right coset Hg of H in G, generated by g, is equal to the left coset gH of H in G, generated by g (where g is an element of G).

ii)

(Still to come).

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# Groupy Theory: Cosets

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