Groups, Subgroups and Normal Subgroups
I'm going over some past papers for an exam but with no answers and have come a cropper on this one, any help appreciated!
Let be a subgroup of a group and let . Define
i) Prove that is a subgroup of .
ii) Let . Which of these statements is true
Give a proof of the statement that is always true
iii) Set . Prove that and that is a subgroup of that is contained in .
Pretty stuck except for ii) is obviously the right one (I hope!).
Re: Groups, subgroups and normal subgroups
(i) and (ii) should be straightforward.
Hint for (iii). Let denote the set of all left cosets of in Each element of induces a permutation of via for each left coset Let The mapping defined by is a homomorphism. Prove that is the kernel of this homomorphism.
Re: Groups, Subgroups and Normal Subgroups
i) let x,y be in Hg.
this means that x = ghg-1 for some h in H, and y = gh'g-1, for some h' in H.
therefore, xy-1 = (ghg-1)(gh'g-1) = ghh'-1g-1,
and hh'-1 is in H (since H is a subgroup), therefore xy-1 is in Hg, so Hg
is a subgroup of G (by the one-step subgroup test).
ii) note that (Hg)k = k(Hg)k-1 = k(gHg-1)k-1 = (kg)H(kg)-1 = Hkg
iii) first, suppose that a,b are in N. this means that a,b are in every gHg-1, for any g in G. by part (i) above, each gHg-1
is a subgroup, so ab-1 is in every gHg-1, so ab-1 lies in N, thus N is a subgroup.
let x be any element of G, and let n be an element of N. note that N = ∩g Hg = ∩xg (Hg)x
since as g runs over all elements of G, so does xg.
but ∩xg (Hg)x = ( ∩g Hg)x = Nx, so N is normal.
(to see that for two subgroups H,K, that (H∩K)g = Hg∩Kg, suppose that
x is in (H∩K)g. then x = ghg-1, where h is in H∩K. hence h is in H, so ghg-1 is in Hg.
but h is in K as well, so ghg-1 is in Kg. hence x = ghg-1 is in both Hg and Kg,
so x is thus in their intersection. on the other hand suppose that x is in Hg and Kg. so
x = ghg-1 = gkg-1, for some h in H, and k in K.
if ghg-1 = gkg-1, then h = k, so h = k lies in H∩K, so x is in (H∩K)g
the same proof holds for any arbitrary intersection of subgroups of G).