plz help me do this, thanks! Plz do the problem by using the given hint, thanks!

Question

plz help me do this, thanks! Plz do the problem by using the given hint, thanks!

In this question we explore the consequences of 2 (i) further. The sets introduced in this question will be very important later in the course. Let G be a finite group and let H be a subgroup of G. For each g G denote by gH the set gH = {gh: h H}. Show that the set gH has the same number as elements as H. (Hint: there are two cases: g H and g H. The question can be answered without consider the cases separately, but I think it could be helpful to do the two cases separately). Show that gH is a subgroup of G if and only if g H. Take any g, k G. Show that gH kH theta if and only if gk-1 H. Show that gH kH theta if and only if gH = gK. (Hint: It might be useful to note that if gk-1 is in H then (gk-1)-1 = kg-1 is also in H, since H is closed under inverses)

Explanation / Answer

I have written the answer on a paper and uploaded it as 3 pictures at the following links:

Part-1: http://i.imgur.com/WuOk361.jpg

Part-2: http://i.imgur.com/pZKGSVH.jpg

Part-3: http://i.imgur.com/80VKWQM.jpg

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