1 3 Let H EG x y or some that is let H be the set all the EG

1 3 Let H EG x y or some that is, let H be the set all the EG: elements of G which have a square root. Prove that H is a subgroup of G

Solution

Solution : let G be an Abelian group.

1) Let H = {x G : x = y² for some y G }

; that is, let H be the set of all the elements of G which have a square root. Prove that H is a subgroup of G.

(i). Let a , b H

, then a = c² and b = d² for some c and d G. The product ab = c²d² shows that ab has a square root because c²d² = ccdd = (cd)(cd) = (cd)². Because G is a group, it is closed under multiplication so cdG.

(ii). Let a H

, then a= c ² for some c G. Since c G,c¹ G because G is a group.