COMEDK2013MathematicsMathematical Reasoning
Which of the following is true?
Options
- AThe set of all fourth roots of unity is a multiplicative group
- BThe set of all cube roots of unity is an additive group
- C(a b)⁻¹=a⁻¹ b⁻¹ for all a, b in any group G
- DIf (a b)²=a² b² for all a, b in any group G , then the group G is non-abelian
Correct answer
A. The set of all fourth roots of unity is a multiplicative group
Step-by-step solution
(a) Set of all fourth roots of unity is aligned &a= 1,-1, i,-i & Now, 1 (-1)=-1 G & 1 i=i G &1 (-i)=-i G &(-1) i=-i G &(-1) (-i)=i G &i (-i)=-i²=1 G aligned So, it is true. (b) Set of all cube root of unit is H= 1, w, w² Now, 1+w=-w² H So, it is false. (c) It is not true, as this is possible only, when a is an abelian group, i.e. a b=b a (d) We have, aligned (a b)² &=a² b² (a b)(a b) &=(a a)(b b) (b)(a) &=(a)(b) b a=a b aligned a is an abelian group. So, it is false.