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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Q: For each of the following groups G, find an external direct product of cyclic groups of prime power…

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Q: -) Write the Cayley table for the group U(15).

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Q: Let A = and B= | of 2 x 2 invertible matrices with real entries un that A and B have finite orders…

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Q: (5) Prove that every group of order 3 is cyclic.

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Q: Show that O(n), the set of all orthogonal n × n matrices is a group under matrix multiplication

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