# Q- (Subgroup Test for Finite Groups). Let G be a finite group. Prove that a nonempty subset H ⊆ G is a subgroup of G if and only if H is closed under the group operation of G.

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Q- (Subgroup Test for Finite Groups). Let G be a finite group. Prove that a nonempty subset H ⊆ G is a subgroup of G if and only if H is closed under the group operation of G.

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Step 1

To verify that H is a subgroup , using the finiteness assumption on G.

Step 2

By definition, a subgroup H of a group G must be (1) closed under multiplication, (2) contain the identity and (3) must be closed under the inverse operation . If G is finite, the  condition (1) ensures that (2) and (3) follow. That is the point of the problem. Note that the statement is not true for infinite groups  For example H=the set of positive integers is closed uner addition , but H is not a subgroup of Z (the additive group of integers. Both (2) and (3) fail to hold in this case,

Step 3

To prove (1) and (2) using the fact that G is f...

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