- #1
wubie
Hello,
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 = D6 = {u, y, y2, x, xy, xy2} where x2 = u, y3 = 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 = Hy2.
ii) Calculate the product AB = (Hy)(Hy2) of the cosets Hy and Hy2 (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 = Hy2 = {uy2, xy2} = {y2, xy2}.
ii)
y*y2 = y3 = u
y*xy2 = xy2
xy*y2 = xy3= xu = x
xy*xy2 = x*xy-1*y2 = u*y = y
Therefore AB = {u, y, x, xy2}
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-1Hg = 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).
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 = D6 = {u, y, y2, x, xy, xy2} where x2 = u, y3 = 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 = Hy2.
ii) Calculate the product AB = (Hy)(Hy2) of the cosets Hy and Hy2 (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 = Hy2 = {uy2, xy2} = {y2, xy2}.
ii)
y*y2 = y3 = u
y*xy2 = xy2
xy*y2 = xy3= xu = x
xy*xy2 = x*xy-1*y2 = u*y = y
Therefore AB = {u, y, x, xy2}
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-1Hg = 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).
Last edited by a moderator: