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Which groups are diffeomorphic?

A. O(2)

B. SU(2,R)

C. Sp(2,R)

D. SL(2,R)

C and D are diffeomorphic.

None of the other groups are diffeomorphic. From the tables on the main page, we see that Sp(2,R) and SL(2,R) are the only two on the list that agree on both compactness and connectedness.

True or False:

The 3 Dimensional image of a Torus is the same as the maximal torus of a matrix group.

False.

A homeomorphism is not the same thing as equality. Even between the flat torus and the "donut" shape, space is distorted.

Does an isomorphism between D_3 and A_3 imply that D_n and A_n are isomorphic?

No.

For example, consider D_4 and A_4. Lower dimensional morphisms do not imply higher dimensional morphisms. In fact, the isomorphism between A_3 and D_3 is called accidental.

Does a matrix group being isomorphic to a torus imply that an element of a matrix group must be conjugate to an element in a torus?

Yes.

An isomorphism defines a mapping between every element so this is implied.