Problem 1.6. In this problem you are going to classify the non-abelian groups of order 8 = 1. Show that every non-abelia

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Problem 1 6 In This Problem You Are Going To Classify The Non Abelian Groups Of Order 8 1 Show That Every Non Abelia 1 (122.13 KiB) Viewed 26 times

Problem 1.6. In this problem you are going to classify the non-abelian groups of order 8 = 1. Show that every non-abelian group of order 8 has an element of order 4. 2. Let G be any group of order 8. Show that if x, y eG are of order 4 then x² = y2. 3. Let G be a non-abelian group of order 8. We know we must have an element of order 4. Now we have two possible cases 2 Case D4 G has an element A of order 4 with an element B EG- (A) of order 2. Show that BAB-1 = A3 and hence G is isomorphic to D4 Case Q G has an element I of order 4 with every element of G – (1) of order 4. Let N = 12. Chose an element J in (1) and set K = IJ. Clearly K EG – (1). (a) Now show 12 = 12 = K2 = N. (b) We must have G = {e, N, I, NI, J, NJ, K, NK} Show JK = I and KI = J (c) Show JI = NK, K J = NI and IK = NJ and hence that G is isomorphic to Q. ==