MATH 403 Study Guide - Midterm Guide: Normal Subgroup, Cyclic Group
10 Jan 2019
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Math 403 - exam 2 - nov. 8, 2004 (15 ) 1. Is even or odd? (c) write as a product of disjoint cycles. (15) 2. If not, show that it is not cyclic. (10) 3. Let g be a nite group with |g| = 10. (a) what are the possible orders of elements of g? (b) prove that every proper subgroup of g is cyclic. Does g have to be cyclic? (15) 4. Let h be the subgroup of s5 de ned by h = { s5 : (3) = 3}. (a) is h a normal subgroup of s5? (you do not need to prove that h is a subgroup. Just check to see if it is normal. ) (b) let , s5. Show that h = h if and only if (3) = (3). (c) let , s5. Try to nd a simple condition similar to that in part (b) above. (45) 5.
