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(Solved): Direct Products and Cyclic Groups (10 Marks) Construct the direct product C_(5)\times C_(2) and list ...



Direct Products and Cyclic Groups (10 Marks) Construct the direct product

C_(5)\times C_(2)

and list all elements. (5 Marks) Define what makes a group cyclic and verify if

Z_(8)

is cyclic by identifying its generators. (5 Marks) 2 Challenging Problems (15 Marks) Using

Z_(2)\times Z_(3)

, illustrate Lagrange's theorem and discuss the order of each subgroup. (5 Marks) For each subgroup, explain why it is or isn't normal within

Z_(2)\times Z_(3)

. (5 Marks) Analyze the group

Z_(15)

and discuss all possible subgroups and their properties, using Lagrange's theorem to discuss the possible orders of these subgroups. (5 Marks)



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