MATH 322 Midterm: MATH 322 2016 Winter Test 1
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Problem 1: let = (cid:18)1 2 3 4 5 6 7. 9 10 (a) write as a product of disjoint cycles. (b) is even or odd? (c) find 999. Problem 2: let g be the group c25 c25 c5. Problem 3: let a and b be elements of a nite group g. show that ab and ba have the same order. Problem 4: let g be a nite group and h be a normal subgroup of g. suppose the index. Show that the commutator subgroup g of g is contained in h. Show that the symmetric group sn does not have a subgroup h of index d, for any 2 < d < n. Problem 6: show that the alternating group a4 does not have a subgroup of order 6. Problem 7: show that every group of order 85 is cyclic. Problem 8: recall that a group g is called simple if g has no normal subgoups, other than.
