A necessary and sufficient condition for a non-empty subset H o f a finite group G to be a subgroup is that

a)
a ∈ H, b ∉ H which implies a, b ∈ H

b)
a ∈ H, b ∈ H ⇒ (a + b) ∈ H

c)
a, b ∈ H ⇒ ab^-1 ∈ H

d)
a ∈ H, h ∈ H ⇒ (a - h) ∈ H

Correct answer is option 'C'. Can you explain this answer?

Question

A necessary and sufficient condition for a non-empty subset H o f a finite group G to be a subgroup is that

a)
a ∈ H, b ∉ H which implies a, b ∈ H
b)
a ∈ H, b ∈ H ⇒ (a + b) ∈ H
c)
a, b ∈ H ⇒ ab^-1 ∈ H
d)
a ∈ H, h ∈ H ⇒ (a - h) ∈ H

Correct answer is option 'C'. Can you explain this answer?

Answer

A) a) H is closed under the group operation of G.
b) H contains the identity element of G.
c) H is closed under taking inverses.
d) H is closed under the group operation of G and contains the identity element of G.
e) H is closed under the group operation of G, contains the identity element of G, and is closed under taking inverses.

The correct answer is e) H is closed under the group operation of G, contains the identity element of G, and is closed under taking inverses. This is because a subgroup must satisfy all three conditions in order to be considered a subgroup.

Answer

Option c is correct Suppose H is a subgroup of G, then H must be closed with respect to composition ∘ in G, i.e. a∈H,b∈H⇒a∘b∈H.

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