Please answer both questions

1. Q+ under multiplication is a proper subgroup of R+ under multiplication.

2.

Theorem A subset H of a group G is a subgroup of G if and only if 1. H is closed under the binary operation of G, 2. the identity element e of G is in H, and 3. for all aH,a1H also. Proof The fact that if HG then Conditions 1, 2, and 3 must hold follows at once from the definition of a subgroup and from the remarks preceding Example 5.11. Conversely, suppose H is a subset of a group G such that Conditions 1,2 , and 3 hold. By 2 we have at once that S2 is satisfied. Also S3 is satisfied by 3 . It remains to check the associative axiom, S1. But surely for all a,b,cH it is true that (ab)c= a(bc) in H, for we may actually view this as an equation in G, where the associative law holds. Hence HG 2. Recall from class that GL(2,R) is the group of invertible 22 matrices with real entries under matrix multiplication. Use Theorem 5.12 to prove that the subset H consisting of all matrices of the form [10a1] is a subgroup of GL(2,R)