test1 - Math 4124 Monday, February 21 First Test. Answer...

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Math 4124 Monday, February 21 First Test. Answer All Problems. Please Give Explanations For Your Answers 1. Prove that Q 8 × S 3 is not isomorphic to D 48 . ( Q 8 denotes the quaternion group of order 8 and D 48 denotes the dihedral group of order 48.) (12 points) 2. Let G be a group. Prove that the formula ( g , h ) · x = gxh - 1 for g , h , x G , defines an action of G × G on G . Show further that if Z ( G ) ± = 1, then there exists a nonidentity element which acts trivially on G (i.e. there exists 1 ± = k G × G such that k · x = x for all x G ). (12 points) 3. Prove that if A and B are subsets of the group G with A B , then C G ( B ) ± C G ( A ) . Is it always true that N G ( B ) ± N G ( A ) ? Justify your answer. (C G and N G denote centralizer and normalizer) (12 points) 4. Let D 16 = ² r , s | r 8 = 1 , s 2 = 1 , rs = sr - 1 ³ , the dihedral group of order 16. Determine for which positive integers n there is a subgroup of order n in D 16 (so you will need to give an example of a subgroup of order
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test1 - Math 4124 Monday, February 21 First Test. Answer...

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