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

ID: 2961244 • Letter: P

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