10/7 2nd Exam
1. Permutation group S9의 한 원소 β가 β^2=(26)(48)(13579)를 만족할 때, β를 구하여라.
2. Show that a permutation with odd order must be an even permutation.
3. Prove that the mapping from U(16) to itself given by x→x^3 is an automorphism . What about x→x^5 and x→x^7? Generalize.
4. Prove that Q, the group of rational numbers under addition, is not isomorphic to a proper subgroup of itself.
5. Let G be a group of order p^n where p is prime. Prove that the center of G can not have order p^(n-1).
6. Let G be a group of order 25. Prove that G is cyclic or g^5=e for all g in G. (e is an identity element of G.)
7. The group S3⊕Z2 is isomorphic to one of the following groups, Z12, Z6⊕Z2, A4, D6. Determine which one by elimination.
8. Show that the group A4 of order 12 has no subgroups of order 6.
