Thread: Groups, Subgroups and Normal Subgroups

Hi there,

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!).

(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.

i) let x,y be in H^g.

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 H^g, so H^g

is a subgroup of G (by the one-step subgroup test).

ii) note that (H^g)^k = k(H^g)k^-1 = k(gHg^-1)k^-1 = (kg)H(kg)^-1 = H^kg

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 H^g = ∩_xg (H^g)^x

since as g runs over all elements of G, so does xg.

but ∩_xg (H^g)^x = ( ∩_g H^g)^x = N^x, so N is normal.

(to see that for two subgroups H,K, that (H∩K)^g = H^g∩K^g, 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 H^g.

but h is in K as well, so ghg^-1 is in K^g. hence x = ghg^-1 is in both H^g and K^g,

so x is thus in their intersection. on the other hand suppose that x is in H^g and K^g. 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).