QUESTION 1. 1) Prove that As has a cyclic subgroups of order 6 and a cyclic subgroup of order 5 but it has no subgroups of order 30. 2) Let A be an abelian group of order 27 such that each element of A different from e has order 3. Up to isomorphism classify all such groups (i.e., up to isomorphism, find all possibilities of such 4) 3) Let A be an abelian group with 75 element. If A has a cyclic subgroup of order 25, then prove that A is cyclic. 4) Let A be an abelian group with 100 elements such that A has no cyclic subgroups of order 25 and it has no cyclic subgroups of order 4. Prove that A has exactly 24 elements each is of order 5. How many elements of order 10 does A have? 5) Let A be a group with 60 elements. Assume that A has a normal subgroup B with 5 elements. Prove that B is the only subgroup of A with 5 elements.