In the following problems, decide if the groups G and G are isomorphic. If they are not, give properties of the two groups that show there can be no isomorphism from G onto G. If they are isomorphic, provide an explicit isomorphism.(a) G = GL(2, R), the group of 2 × 2 nonsingular matrices under multiplication; G = (R − 0, ·), the nonzero real numbers under multiplication.(b) G = (R, +), the real numbers under addition; G = (Q, +), the rational numbers under addition(c) G = Q4, the group of quaternions; G = D4, the dihedral group of symmetries of the square

Question
Asked Sep 10, 2019

In the following problems, decide if the groups G and G are isomorphic. If they are not, give properties of the two groups that show there can be no isomorphism from G onto G. If they are isomorphic, provide an explicit isomorphism.

(a) G = GL(2, R), the group of 2 × 2 nonsingular matrices under multiplication; G = (R − 0, ·), the nonzero real numbers under multiplication.

(b) G = (R, +), the real numbers under addition; G = (Q, +), the rational numbers under addition

(c) G = Q4, the group of quaternions; G = D4, the dihedral group of symmetries of the square

 
 
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Expert Answer

Step 1

(a)   We are given that G = GL(2, R), the group of 2 × 2 non-singular matrices under multiplication; G= (R − 0, ·), the nonzero real numbers under multiplication. We need to check whether the groups G and G are isomorphic or not.

Note:

Two groups are isomorphic means they are exactly the same except for the names of the elements and the name of the binary operation in both of the groups. An isomorphism between two groups is a function that renames all of their elements.

Here, it can be observed that the group G is an abelian group whereas G is a non-abelian group, which implies that both of the groups are having different properties.

Step 2

Check:

Let φ : G à G , where φ(A*B) = φ(A) × φ(B).

(  here A, B ∈ G, * represents the matrix multiplication, φ(A), φ(B) ∈ G  and  × represents multiplication of two non-zero real numbers.)

Since,  G is a non-abelian group, φ(A*B) ≠ φ(B*A)

                                                      =>  φ(A) × φ(B) ≠ φ(B) × φ(B), which is a contradiction.

Hence, G and G are not isomorphic to each other.

Step 3

(b)  We are given that G = (R, +), the real numbers under addition; G= (Q, +), the rational numbers under addition. We need to check whether the groups G and G are isomorphic or not.

Note:

If two groups are isomorphic to each other then the cardinality of both of the groups must be same. If not then ...

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