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)