# ans2 - Math 312 Assignment#2 Due on Wed 7 October 2015...

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Math 312Assignment #2Due on Wed 7 October 2015Solutions should be correct, complete, clear, and neatly presented to get full marks.It is possible that only a proper subset of the following questions will be marked.(1)(a) Find the number of elements of order 9 inZ9Z9.One way to do this is to count the number of elements of order<9. These arethe ordered pairs (a, b) such that bothaandbcome from{0,3,6}. The numberof these is 9 and so the number of elements of order 9 is 81-9 = 72.(b) Find the number of cyclic subgroups ofZ9Z9of order 9.Every cyclic group of order 9 has 6 generators, so the number is726= 12.(2) Determine ifZ60Z6Z2is isomorphic toZ10Z6Z12.Yes. The first groups is isomorphic to (Z5Z4Z3)(Z3Z2)Z2and the secondone is isomorphic to (Z5Z2)(Z3Z2)(Z3Z4)(3) Is the groupZ4Z2an internal direct sum of two nontrivial groups other than theobvious ones:Z4⊕ {0}and{0} ⊕Z2?You look for two subgroups that satisfy the conditions of internal direct productand also (by the uniqueness part of the Fundamental Theorem of Finite AbelianGroups) must also by cyclic of order 4 and 2.