(a).Let me show that H is a subgroup. Then you can make the identical argument and show that K is. e = (e, e) is in H let (x, e) be in H. G1 group so x^-1 in G1. Now (x^-1, e) = (x,e)^-1 since (x, e)(x^-1, e) = (xx^- 1, e) = (e, e) which is the identity in G let (x, e), (y, e) be in H. Then (x,e)(y,e) = (xy,ee) = (xy,e) is in H. Therefore H is a subgroup of G b) Let me show that KH=G. You can make the identical argument to show that HK=G. Let k =(e, y) be in K, h=(x, e) in H. Then kh = (e,y)(x,e) = (ex,ye)=(x,y) is in G, so KH subset of G Let (x,y) be in G. Then (e,y) is in K and (x,e) is in H, and (e,y)(x,e) = (x,y). So G is a subset of KH. Therefore G=KH Solution (a).Let me show that H is a subgroup. Then you can make the identical argument and show that K is. e = (e, e) is in H let (x, e) be in H. G1 group so x^-1 in G1. Now (x^-1, e) = (x,e)^-1 since (x, e)(x^-1, e) = (xx^- 1, e) = (e, e) which is the identity in G let (x, e), (y, e) be in H. Then (x,e)(y,e) = (xy,ee) = (xy,e) is in H. Therefore H is a subgroup of G b) Let me show that KH=G. You can make the identical argument to show that HK=G. Let k =(e, y) be in K, h=(x, e) in H. Then kh = (e,y)(x,e) = (ex,ye)=(x,y) is in G, so KH subset of G Let (x,y) be in G. Then (e,y) is in K and (x,e) is in H, and (e,y)(x,e) = (x,y). So G is a subset of KH. Therefore G=KH.