Question

2.

problem 3.

Let H be a normal subgroup of a group G and let K be any subgroup of G. Prove that the subset HK of G defined by is a subgroup of G Let G S, H ), (12) (34), (13) (24), (1 4) (23)J, and K ((13)). We know that H is a normal subgroup of S, so HK is a subgroup of S4 by Problem 2. (a) Calculate HK (b) To which familiar group is HK equal? (c) Is HK a normal subgroup of S. Justify your answer.

Answer 1

(2) Given :H is a normal subgroup of G. K is a subgroup of G Proof: Assune x, v eHK x_hk and v-hk, for some h.h EHand kkEK Since HK KH kh=h,k, for some h、EHand k、EK ry e HK >HK is closed under multiplication r(hk) -1 (k)-1 (h)-1 KH Since KH -HK, e HK HK is a subgroup of G.