Q67,paper3, J12. Let a * H and b * H be two cosets of H.

(i) Either a * H and b * H are disjoint
(ii) a * H and b * H are identical
Then,
(A) only (i) is true
(B) only (ii) is true
(C) (i) or (ii) is true
(D) (i) and (ii) is false
Answer©.
Explanation
If H is not normal in G, then its left cosets are different from its right cosets. That is, there is an a in G such that no element b satisfies aH = Hb. This means that the partition of G into the left cosets of H is a different partition than the partition of G into right cosets of H. (Some cosets may coincide. For example, if a is in the center of G, then aH = Ha.) On the other hand, the subgroup N is normal if and only if gN = Ng for all g in G. In this case, the set of all cosets form a group called the quotient group G / N with the operation ∗ defined by (aN ) ∗ (bN ) = abN. Since every right coset is a left coset, there is no need to distinguish "left cosets" from "right cosets".