Question

8. Let H be a subgroup of a group G and let X be the set of left cosets H of H in G. (i) Show that g.xH ::: grH (g, x є G) defines an action of G on X. (ii) Show that there is a single orbit, X, under this action (ii) Show that the stabilizer of the coset rH is Hx (iv) Show that the kernel of the action is a normal subgroup of G that is contained in H 8 marks]

I have done the first part showing that it is an action, can anyone help the rest of them please! Many thanks!!!

Answer 1

) We have to show that there ie a single orbit、 for x under this acton. That means we have bo show that any two enosen elemen same under thie achion Chiven action is^, fer g.xe* Then, we need to show there erist qe such that So, choose 1-y.ґ Then we have gan-.yxaリ Mence zH and g» are equivilent under tie yH are equiwa adioh Since, theg were chosen ascarty a

the elements ef xH are cquivalent Hence there exxist a stnale orbit for X , under the given ach'on ii) "The stabiliser of H is the XLS ne stabiliser ot x H

w) -The kernel of the achion is given by ed Fürst we wall pove the kernel is contain Let g k Hence EH S, gek get Hence Kernel k is contained n H. in h

Nlow we vwill show that K ts a subgroup of Ch Let 3,3 e k Hence k is cased ouw ideni e belong to k since Nlow let qek then

So invaise cxist for all gek So) K is a subgroup of Now we have to show k is normal in i For that s enough bo chow k cs abelian for all xH EX Kis a normal subgroup ofん(n