Question

19. A non empty subset H of a group (G, *) is said to be a subgroup if
(A) HCG and (H, *) is a group
(B) Commutative and associative laws hold in H
(C) HCG
(D) None of these

Answer 1

Thus by closure, a b^-1 ∈ H.

Option (A) is correct.

Explanation:

• Let G be a group. Show that a nonempty subset H is a subgroup of G if any only if ab−1∈H for any a,b∈H.
• The forward direction is quite easy.
• Suppose H is a subgroup.
• Then by closure, ab∈H for any a,b∈H.
• Every element has an inverse.
• Therefore if b ∈ H, then b^−1 ∈ H.
• Thus by closure, a b^-1 ∈ H.