(a) Set if all fourth roots of unity is a={1,−1,i,−i} Now, 1×i=i∊G 1×(−i)=−i∊G (−1)×i=−i∊G (−1)×(−i)=i∊G i×(−i)=−i2=1∊G So, it is true. (b) Set of all cube root of unit is H={1,w,w2} Now, 1+w=−w2∉H S, it is false. (c) It is not true, as this is possible only, when a is an abelian group, i.e. ab=ba (d) We have, (ab)2=a2b2 ⇒(ab)(ab)=(aa)(bb) ⇒(b)(a)=(a)(b)⇒ba=ab ⇒a is an abelian group. So, it is false.