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Show that the center of GL(2,R) is the set of all scalar matrices aI with a different from zero.

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Problem 1: Show that the center of GL(2,R) is the set of all scalar matrices aI with a different from zero.

Problem 2: Prove that no pair of the following groups of order 8,
I8; I4 x I2; I2 x I2; D8; Q, are isomorphic.

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Matrix and grouptheory problems are solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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Problem #1
Proof:
Suppose is the center of the group . We consider an element with the form . Then commutes with any element in . Then we have

Then we have , . So we have .
We also have

Then we have
Thus . Since is invertible, then .
It is easy to verify that commutes with any element in ...

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